Introduction - Pedagogická fakulta MU This textbook was ... facts themselves, examples of possible examples that may be used to present the ... During archaeological excaationsv in

Introduction

This textbook was written for students of pedagogical faculty for all branches ofstudy. It is divided into two parts: its �rst part gives a short overview of thehistory of mathematics from antiquity up to the most recent developments. It isobvious that the scope of this textbook does not allow for a detailed overview ofthe history, and therefore the author focused on those parts of history of math-ematics that may be used at elementary and secondary schools. Apart from thefacts themselves, examples of possible examples that may be used to present thehistorical material in schools are given. Second part contains short biographies ofsigni�cant mathematicians, where preference was again given to those with whosenames pupils are likely to meet during their mathematics lessons.

1

2

Part I

Selected parts from history of

mathematics

3

Chapter 1

Mathematics of the antiquity

While historians who are mapping the development of mathematics in, say, thesecond half of 19th century have a relatively easy task, because it was then commonfor researchers to publish their results and these publications have been preservedin libraries, historians researching the development of mathematics face a verydi�cult task, as practically no sources have been preserved, had there ever beena su�cient amount of them. In any case, there is evidence that even peoplein Paleolite had mathematical knowledge. During archaeological excavations inDolní V¥stonice, the world-famous Moravian researcher, Professor Karel Absolon,found among other things also a wolf's rib bone with 55 notches, with each �fthnotch longer than the others. Later, similar bones were discovered at other places.Bones that are in this way changed (bones with notches) are nowadays called tallysticks. Although there are still discussions going on as to what these bones wereused for, it is very probable that they served for counting objects. After all, mostcontemporary waiters behave in a similar manner.

For contemporary youth, the books of adventures from Palaeolithic periodwritten by Eduard �torch, which were very popular among boys when I was young,might now be not as popular, but I will nevertheless remind the reader of one ofthem. The hero of the book Mamoth Hunters is a teenager called Kop£em, verythoughtful and very educated for his time, because he could count up to �ve, whichnobody else, apart from the chief and a handful of top hunters, could do. When thenumber of objects was greater than �ve, it was said that there were many. Thus,in some sense, our ℵ0 was known already to mamoth hunters, only their countablesets were all sets with the number of elements higher than �ve. It is interesting tonotice that some natural nations, untouched by Euro-Atlantic civilisation, countin the same way until today. A similar example may be found in current naturallanguages: in Czech, we count jedna, dv¥, t°i, £ty°i, etc. However, when we speakof a certain part of the whole, we have polovina, t°etina, £tvrtina, etc. A similarcuriosity may be found in other languages as well, for example the English countone, two, three, four and divide into halves.

While the Palaeolithic and Mesolithic man was a nomad, hunter, and picker,people in Neolith started to settle at one place, at least for a certain period of

5

6 CHAPTER 1. MATHEMATICS OF THE ANTIQUITY

time, since their way of obtaining food had changed. From nomads and hunters,people became a settled farmer, people started building permanent residences, andhuman society began to break up into atoms, as people began to concentrate ona certain mode of working. First two great divisions of labour take place: cul-tivation of land was separated from shepherding and agriculture from the crafts.Construction of homes, production of tools and ceramics, all that suggests thatmathematical, and especially geometrical, knowledge of people was being devel-oped. As can be shown in archaeological �ndings, people decorated ceramics withgeometrical patterns, and historians even denote the individual cultures throughits ornaments, because we cannot know what these peoples were called (peoplewith spiral ceramics, voluta ceramics, culture of bell-like goblets, etc.). With theaccumulation of material wealth, the society could a�ord something that had be-fore been unthinkable. Some people were excluded from the production processand devoted their time to spiritual activities. This, however, brings us to the pe-riod that is traditionally called antiquity and which is characterised by a changein the organization of the society, we speak of civilizations.

1.1 Mathematics of Ancient Egypt

Egyptian civilization originated at the lower reaches of the Nile, the origins ofthe Egyptian state being assumed to be around 3000 BC, when Upper and LowerEgypt were connected. The Egyptian empire founded by Meni can truly be calleda thousand-year empire, because it lasted without interruption until 525 BC, whenit was brought down by the Persians. In order to help us imagine how long thisempire lasted, let us compare it with this country. If we deduct from today's date2500 years, we reach the year 600 BC. Historians do not even know what the namesof the tribes inhabiting this area were; Celtic tribes of Boja only came a centurylater. Renaissance of Egypt was taken care of by one of the military leaders ofAlexander the Great, Ptolemaios, who founded the last royal dynasty of AncientEgypt.

Ptolemaios empire, especially during the reign of the �rst Ptolemaians, pros-pered and became the centre of education of the contemporary world. In the newlyfounded capital of this country, Alexandria, Múseion (the House of the Muses)was founded by Démetrios from Faleras and in this institution, almost all of thegreatest scientists of that time were working. Even the best known mathematicaltreatise of that time, Euclid's Elements, were written in this institution, generouslyfunded by the king-farao Ptolemaios I. The name of this institution is preserveduntil today in the word museum, but the activity of this institution rather resem-bled the activities of our contemporary Academy of the Sciences. Over a shortperiod of time, they also succeeded in building a truly large library in Alexandria,comprising hundreds of thousands of scrolls (rolls of parchment). It is said that ifsomebody came to Alexandria and carried a book them, this book was taken fromthem and deposited in the library, while the original owner had to be content witha copy only.

1.1. MATHEMATICS OF ANCIENT EGYPT 7

After the death of the last Queen of Egypt, Cleopatra VII,1 Egypt became partof the Roman Empire. In the 7th century AD Egypt was conquered by the Arabsand they managed to �nish what had been begun already by Caesar and in whichthe Christian Roman emperors successfully continued, led by the pious Theodosios,namely to destroy the remnants of the once famous Alexandrian library. Thetradition has it that when Arab soldiers asked their military leaders what to dowith the books, the answer was, "Burn them. Either they say something that isin Koran, and then they are useless, or they say something else, and then they areharmful."2 In order to provide a balanced picture, we must say that such attitudewas rather exceptional and held only for the early period of Islam. In the areasthey conquered, science was well tended, however, the libraries founded by themwere later a welcome object for vandalism, this time conducted by Christians.

For modern Europe, Egypt was discovered by a Napoleonic expedition towardsthe end of 18th century, which was not successful for France in the military sense,but was very important from the point of view of getting to know the cultureof Ancient Egypt. Bonaparte namely brought with him also respectful Frenchscientists, from mathematicians, Monge and Fourier took part in the expedition,among others. Although the French wanted to ship the �ndings back home inorder to protect them from the uneducated fellahs,3, the Europe of that time got toknow about Ancient Egypt thanks to them and a new branch of science originated -Egyptology. By the way, Napoleon Bonaparte was one of the few modern monarchswho realised the importance of science for the society. Tradition has it that beforethe battle, he always commanded scientists and donkeys forward. When we realisehow important donkeys were for the army, especially for transferring load, thiscommand is not pejorative for the scientists, quite the contrary. This artilleryo�cer and later emperor was interested in the most recent scienti�c discoveriesand when prizes were awarded, scientist were not crouching at the back, but on thecontrary, were sitting in the front rows. He himself was well versed in mathematicsand could appreciate its importance.

Let us, however, return to Ancient Egypt. Although the Egyptians are said tobe the most diligently writing people of the Ancient times and although they leftus many written sources, unfortunately only very few mathematical texts havesurvived until the present time, perhaps because mathematical texts were notcarved out in hieroglyphs into stone, but were written in hieratic script on papyrirolls. The preserved monuments, especially the pyramids, tombs, and church com-plexes, prove that mathematical knowledge of the Egyptians was wide-reaching,even though probably rather empirical in character.

The best known Ancient Egyptian mathematical text is the so-called RhindPapyrus, which was written in 16th century BC, but whose draft is roughly threecenturies older. It was named after the English antique dealer Rhind who boughtit in 1858. Two of its large fragments are now in British Museum in London,

1This lover of the Romans Ceasar and Marcus Antonius is the main character of several novels,dramas and �lms and is apparently the best known Egyptian woman ever.

2The same story is passed around about the conqueror of Central Asia.3In the end, they had to be content only with plaster casts and drawings ot the outstanding

artist Denon; the originals were taken by the victorious English.


several smaller fractions have found their way to the Brooklyn Museum. It iscomposed of 14 sheets 38-40 centimetres wide and the length of the two biggerfragments added together is 513 centimetres. Moscow Papyrus was bought by theRussian Golenishchev at the end of 19th century and we will nowadays �nd it inPushkin Museum of Fine Arts in Moscow. Its original length and width are 544and 8 centimetres, but today, it comprises of one larger piece of 11 sheets and ninesmall fractions. Several fractions of a scroll with mathematical text were foundin Káhún in 1889, two small fractions are owned by The Egyptian Museum inBerlin. In Egyptian Museum in Cairo, they take care of two wooden tablets withmathematical texts and by those, we have exhausted the list of Ancient Egyptianmathematical sources.

Let us now look what mathematical knowledge of the Ancient Egyptians hasbeen preserved until today. The basic element of mathematics is a number andAncient Egyptians worked with whole numbers and with fractions. For wholenumbers, they developed a decimal non-positional notation, using digits from 1 to1,000,000. Numbers were denoted through combinations of the necessary amountof digits for units, tens, hundreds, etc. In Egyptian way, the number 42 would bewritten in the following way: as one, one, ten, ten, ten, ten. Egyptians only usedunit fractions, i.e. the reciprocals (multiplicative inverses) of natural numbers.Counting with fractions was thus rather tedious, because the result had to be aunit fraction again. In order to make the job easier, they designed tables in whichfractions were expressed as a sum of several unit fractions, e.g. 2

3 = 12 + 1

6 .Ancient Egyptians knew basic arithmetic operations, multiplication and divi-

sion being based on addition. One factor (multiplicand) was doubled as manytimes as it took to reach the second factor (multiplicand) and the relevant resultswere then added up. So should an Ancient Egyptian multiply 51 by 11 and didnot have a calculator at hand, they would proceed in this way: 1x51 = 51, 2x51= 102, 4x51 = 204, 8x51 = 408. If we add up the items 1, 2, and 8, we receivethe correct result 561. On the other hand, in order to divide one of these numbersby the other, we double the divisor until we receive the dividend. 51 = 44 + 4 +2 + 1, therefore 51:11 = 4 + 4/11 + 2/11 + 1/11. Raising a number to a powerand extracting a root of a number is not very common in Ancient Egyptian texts,it is mostly performed on "decent" numbers, and it is probably that geometricalinterpretations were used to perform these operations. Great care was devoted tocalculation with unit fractions, but since detailed description of these operationsexceeds the extent of this textbook, we refer the reader to other books for thetreatment of this matter, for example [Vy]. As Ancient Egyptians solved solelypractical problems, the computers had to be able to translate between the di�erentunits of measure.

We may deduce from the preserved mathematical texts that Ancient Egyptianshad comparatively extensive knowledge of geometry, especially when it comes toits practical uses, i.e. mainly the calculations of areas and volumes. Ancient Egyp-tians could calculate the area of a rectangle. They only knew isosceles trianglesand their areas were computed as the area of the rectangle with an equal area.The area of a trapezium was calculated by transforming the task to calculating thearea of an isosceles triangle whose base was as long as the sum of the lengths of the

1.1. MATHEMATICS OF ANCIENT EGYPT 9

two bases of the trapezium, in other words, they knew the formula S = (a+c).v2 .

A circle was determined by its radius and its area was calculated by transformingit to a square of a roughly equivalent area. The length of the side of the squareemerging from this transformation and the diameter of the circle were in the ratio89 . Ancient Egyptians did not know the number π, but their results correspond toπ.= 3.16.As the pyramids, the only of the seven wonders of the Ancient World that has

survived until these days and which probably also our descendants will be ableto admire, Ancient Egyptians also had ver good knowledge of stereometry. Theycould calculate the volumes of a rectangular cuboid and a roller, in which theyproceeded as we do nowadays, namely the area of the base is multiplied by theheight. They could also calculate the volume of a truncated pyramid, but it isinteresting that they had no special word for this solid and they only depicted itthrough in an ideogram. We could describe their procedure through he formulaV = 1

3v(a2 +ab+ b2). In Rhind papyrus, we can even �nd problems leading to thecalculations of the slant angle of the pyramid. We will not �nd the substantiation ofthe procedure in the stereometry problems either and in contrast to the planimetryproblems, not even a uni�ed terminology was preserved. As the Ancient Egyptianmathematical documents originated relatively late, it is probable that their writersmade use of the thousands of years of experience of Ancient Egyptian engineers.

In the preserved mathematical texts, tasks leading to arithmetic and geometricsequences may be found, among others tasks concerned with comparing the qualityof beer and bread. Some tasks lead to solving equations and in some, the solution isfound thanks to the method of false position. A small sample of Ancient Egyptiantasks is in the text below.

A triangle whose height is ten and base four. Tell me its area. Calculate ahalf of four, that is two, to determine the rectangle. Calculate with ten twice, theoutcome is twenty. That is its area (Moscow papyrus, problem 4)

Method of calculating the area of a circle with (the diameter) 9 chet. What is itsarea? Subtract one ninth of it, that is one, and the remainder is eight. Calculatewith eight eight times, you obtain sixty-four. That is its area on the surface. 64secat. (Rhind papyrus, problem 50)

Method of calculating a sack with many precious metals. When it is said: asack in which there is gold, silver, and pewter. This sack can be obtained for 84shatei. What it is that corresponds to each metal, if a deben of gold gives 12 shatei,(for) silver it is 6 shatei and (for ) pewter it is 3 shatei. Add up all that is givenfor a shatei of all metals, you obtain 21. Calculate with those 21 until you get 84shatei. That is what you can get the sack for. You obtain 4. That is what yougive for each metal. Procedure: calculate with four twelve times, you obtain: goldis 48, that is what belongs to it. 6x silver 24, 3x pewter 12. 21 altogether 84.(Rhind papyrus, problem 62)

Method of calculating 16 measures of Upper Egypt barley. Convert 100 loaves(of quality) 20, the rest for beer (quality) 2, 4, 6 1

214 malt for dates. Calculate the

share of the bread (of quality) 20, you obtain 5. Calculate the remainder from16 for 5, you obtain 11. Do the division by 1 for these qualities obtaining 2

314 .

Calculate with 2314 twice, because it was said 1

216 malt for dates, you obtain 1 2

316 ..


Calculate with these 1 2316 . until you �nd 11, obtaining 12x. Tell him: this is the

corresponding beer. You found out correctly. (Moscow papyrus, problem 13)Let us add a short commentary to the last problem. It is obvious that for 100

loaves of bread of quality twenty, we will use 5 heqats of barley. For a jug of beerof any quality, we will use 1

2 + 14 + 1

6 of heqats of barley, which, after adding thedates, adds up to 11

6 heqats. By dividing 11 heqats by this value, we will get thenumber of beer jugs. A careful reader has surely noticed that there are not twelvejugs, but only six. It is thus quite possible that the author of this problem likedbeer a lot and in this case, it could have been wishful thinking.

1.2 Babylonian mathematics

Another area where ancient civilizations developed was Mesopotamia that is thearea between the rivers Euphrates and Tigris. In contrast to Egypt, this areawas inhabited by several nations in a sequence (Sumerians, Akkadians, Babyloni-ans, Assyrians); to make things simple, we will talk about Babylonian mathemat-ics, although this denomination is not quite accurate. In this area, the so-calledcuneiform4 was used, it was written on clay tablets, which were �red (in kilns),so a relative plenitude of sources for these empires was preserved. It seems thatBabylonian mathematics was more advanced than Ancient Egyptian mathematics,at least this is what the preserved sources show. One of the greatest contributionsof Babylonian mathematics was the use of positional notation for numbers. Baby-lonians used number sixty as the base, perhaps because the number 60 has manydivisors. The division of the circle into 360 degrees and of the hour into 60 minutesand the like have their origin in Babylonian mathematics. These divisions are sostrongly anchored in human minds that not even the creators of SI units, which isotherwise strictly based on decimal number system and its units are, apart fromthe exceptions, multiplied by a thousand or divided into a thousand parts, daredto divide an hour into a thousand of mili-hours and they only applied this divisionin the smallest unit, namely the seconds.

A hexadecimal number system was used already by the oldest people whoinhabited Mesopotamia, namely the Sumerians. While for us, number 1011 isexpressed as 1.103 + 0.102 + 1.101 + 1.100, sometimes without being fully aware ofthis, the Sumerians apprehended this number as 16.601 + 51.600. The Sumerians,however, did not know the zero, not even as a sign for an empty space. Thepeoples who came to the area between Euphrates and Tigris used the followedtheir example. The computer thus had to realise from the context of the problemwhich number the notation corresponds with. It was only towards the end of theBabylonian era that numbers with a sign for the zero appeared at the empty space.The Babylonians were excellent computers; the tablet YBC 7289 from the era ofthe famous king Hammurabi (around 1700 BC) contains a calculation of

√2 with

precision to one millionth. To make the calculations easier, they used, like theBabylonians, various tables. For example, Plimpton tablet number 322 contains a

4Cuneiform is the script that has been used for the longest period of time: it was used formore than three thousand years

1.2. BABYLONIAN MATHEMATICS 11

number of Pythagorean triples.The Babylonians were also experts in solving individual equations and their

systems. Solving linear equations was considered trivial, but they also dared tosolve quadratic equations and some types of cubic equations. As they did notknow notation, they had to describe the individual operations in words. In thesame vein, they did not consider negative numbers to be numbers, and thus theequations had to be given in such a way that they only contained positive numbersand negative solutions were ignored. We will demonstrate this on solving theequation x2 +x = 0.75. The left side of the equation was completed to the square,namely (x + 0, 5)2 = 1, thus x + 0.5 = 1, therefore a x = 0, 5. The other root,x = −1.5, is negative, but they were not looking for it, although it could easily befound this way (x+ 0.5 = −1⇒ x = −1.5).

An interesting problem can be found in tablet VAT 8398. From (1) bur Iharvested (4) gur of grain. From one other bur I harvested (3) gur of grain. Grainexceeds grain by (8.20). My �eld added and it gives (30.0). What are my �elds?This problem is written in a rather complicated way, it contains various units,which the Babylonians could transfer into another and back (in contrast to us),and hexadecimal notation is used for the numbers. To understand the Babylonianprocedure better, we will formulate the problem in another way. A box of bottlesof beer was brought to the bricklayers. Kvasars cost CZK 15 per bottle, VelensCZK 12 per bottle. We paid CZK 276 altogether (without deposit). How many ofeach there are? A Babylonian would have assumed that there were equal numbersof bottles of each kind, and then we would pay 10.15 + 10.12 = 270 (CZK), soit is clear that there must have been more Kvasars than Velens, and he wouldhave realised that by changing a Velen for a Kvasar would increase the price ofthe purchase by 3 crowns. It thus su�ces to divide the di�erence between thecalculated price and the given price by 3 and we immediately know that therewere 12 Kvasars and 8 Velens. Today we solve similar problems with the help ofa system of two equations with two unknowns, but according to the author, themethod of false position is the most natural way of solving such problems and itis a pity that it is not mentioned in current textbooks. It would namely be noproblem to substitute the beer sorts with lemonades, which are also produced in�erná Hora brewery, or with another produce suitable for children, or, in orderto increase the awareness of prices of various goods, transform the problems intosimilar ones with contemporary units and prices.

Babylonians knew the formula for calculating the square of a binomial, thedi�erence of two squares, the sum of �rst n members of a geometrical progression(geometrical sequence) with ratio equal to two, and the sum of the �rst n squaresof natural numbers, although they only had at their disposal the description ofthe procedure in words and expressing it in a formula would probably be moreprecise. The above-mentioned Plimpton tablet proves that they could constructPythagorean triples and that they were also aware of the geometrical interpretationof these numbers. As far as knowledge of geometry is concerned, however, theywere slightly behind the Egyptians, or at least the currently known �nding suggestthat they could determine the area of a triangle and a trapezoid and that theyknew approximate procedures for calculating the volume of a roller, a cone, and of


a polygonal frustum. Peoples living in Mesopotamia, however, constructed theirbuildings with �red bricks, which were not preserved until today, except for thefoundations.

Chapter 2

Mathematics in Ancient

Greece

The Greek tribes, in several waves, settled not only in the area of contemporaryGreece, but also the islands in the Aegean Sea and Anatolia (Asia Minor). Itis precisely in the towns of Asia minor where we can �nd the origins of Greek,and thus also European, science. It is only since that time that we may speakof mathematics, Greeks were the �rst ones who started doing mathematics in thesense we know it. While before, only speci�c tasks were solved, the Greeks beganto construct mathematics as a branch of abstract science, they began to generalizetheir �ndings and prove them. The mathematics of Ancient Egypt and Babyloniais for us practically anonymous, although we know some names, but in the caseof Ancient Greece, we know the names of men whom we may call mathematiciansin the contemporary meaning of the word. And these names are known even topeople who have nothing to do with mathematics. The names of Archimedes,Euclid, Pythagoras, Thales, and others are generally known today. Their livesand especially work will be mentioned in particular in the second part of thistextbook, but the results of their mathematical work cannot be missing from anyshort summary of the development of mathematics.

2.1 Figurate numbers

We will probably never learn why in Ancient Greece the face of mathematicschanged so signi�cantly from concrete problem-solving to an abstract science. Ac-cording to some researchers, �gurate numbers were one of the things that couldhave contributed to this. A natural number may namely be visualised in multipleways - and the ensuing con�gurations are called �gurate numbers (triangle, rectan-gle, square, pentagonal, etc.). Maybe it was a former soldier or physical educationteacher who embarked upon the idea to order pebbles into couples and found outthat sometimes, the resulting shape is closed (even numbers), and sometimes not(odd numbers). We will obtain the same result if we form one set of couples from

13

14 CHAPTER 2. MATHEMATICS IN ANCIENT GREECE

two such sets, while it is irrelevant how large the original shapes were, but solelyon the fact whether the original shapes were closed or not. This way, the �rstmathematical theorem was born and proved at the same time. On the other hand,I am not sure whether this �nding was important for the working people, be it anarchitect or a tradesman with olives.

Triangular �gural number can be expressed by the formula a = 12n(n + 1),

which is also the formula for determining the sum of the �rst n members of anarithmetic series, in which the �rst member is equal to 1 and the di�erence as well(i.e. a1 = 1 and d = 1). Square �gural numbers are determined by the formulaa = n2. When we look at their practical construction, we can see that the �rstsuch number is determined by one pebble. In order to get the next number, wehave to add two pebbles in the horizontal direction and one in the vertical one,i.e. three in total. When constructing the third of one of square numbers, weadd three pebbles in the horizontal direction and two in the vertical one, i.e. �vealtogether. This procedure is the visualization of the formula

1 + 3 + 5 + · · ·+ 2n− 1 = n2

or, expressed in words, the sum of n odd numbers is equal to n2.

2.2 Commensurability and the discovery of irratio-

nal numbers

From Ancient Greek legends, we may deduce that the Hellenic people were fondof music, and perhaps it was music which inspired Pythagoras and his pupils tocreate their special philosophical theory. Pythagoreans namely noticed that thetones formed by two strings sound melodiously precisely in those cases when theratio of their lengths can be expressed in small integers. If we press the secondstring exactly at the half, we will get the octave (2 : 1), by shortening the stringto two thirds, we will get the �fth (3 : 2), the fourth is represented by the ratio (4: 3), etc. Why, then, should we look for the essence of the world in the �re, thewater, or the inde�nite apeiron, when we have the number. The essence of beingis the number (arithmos), only the number will enable us to describe quantitativerelationships of objects and phenomena. And just like the music will be melodiouswhen the lengths of the strings are in the ration of small integers, the world willalso be beautiful if we are able to express it in a similar way.

According to their imagination, there was the central �re in the middle of theuniverse, around which the Earth, the Sun, the Moon, the individual planets, andthe stars rotate. As the Greeks only knew of �ve planets, it was necessary toinvent also the Anti-Earth, so that there were ten spheres on which the heavenlybodies circulated. For Pythagoreans, the ten was a symbol of perfection, and thusthey could not stand it if there were only nine spheres. The spheres were naturallyperfect spheres, their radii were in ratios of small integers, and the motion of thecelestial bodies was uniform. Just like music originates through the motion ofthe string or through vibration of the air column, there must have been musicoriginating from the movement of celestial bodies, according to the Pythagorean

2.2. COMMENSURABILITY AND THE DISCOVERYOF IRRATIONAL NUMBERS15

imagination, beautiful, perfect music, which could only be perceived by greatminds, since this kind of music can only be perceived through the reason. TheGreek word kosmeo means beautiful, therefore the name cosmos and the wordcosmetics are derived from the same root.

One was not regarded as a number, but as a basic building element of arith-metic. Natural numbers were collections of a certain number of units; even num-bers were female, odd numbers male. Rational numbers were represented as ratiosof natural numbers. Every two numbers were commensurable, because there al-ways was the smallest number dividing both the original numbers. This happystate, however, did not last long, because irrationality (incommensurability) wasdiscovered and the Pythagorean model of the world collapsed. It was prohibitedto reveal this discovery to the public, but one cannot guide such secret forever,and soon also the people outside the community learnt about this discovery. Hip-pas from Metapont, who was said to have revealed the discovery, was even killedaccording to some legend, according to others, he killed himself, and according toyet others, he was punished by the gods.

The proof that the root of 2 is irrational belongs today to school examples ofthe application of proof by contradiction. If

√2 was a rational number, it would

be possible to write it down as a fraction, namely√

2 = pq , for (p, q) = 1 (i.e.

the greatest common divisor of the two numbers a and b is 1). After squaringboth sides of the equation and reduction, we get p2 = 2q2, which means that p2

is even, therefore also p is even, and can therefore be written as p = 2k, and aftersubstituting it into the original equation, we deduce that q is also even, which is incontradiction with the assumption that p and q are mutually prime. This proof issimple, but the Greeks have discovered the incommensurability of

√2 di�erently;

we present the two most common possibilities in the text below.Let a square be given, whose side is a units long and the diagonal u units long.

At least one of these numbers is odd, because otherwise we could double the unit.According to the Pythagoras theorem, u2 = a2 + a2, hence u2 is also even, andthus also a half of the diagonal can be measure with an integer number of units.Through another use of the Pythagoras theorem, we get a2 =

(u2

)2+(u2

)2, which

means that a2 is also even, which is in contradiction with the assumption. Thisproof is essentially a geometric modi�cation of the proof given above.

Pythagoreans liked a lot the regular pentagon, and ascribed magical power toit. If we are to construct a regular pentagonABCDE, it su�ces to construct anisosceles triangle ABC and the remaining points are found easily. As accordingto the Pythagoreans, every two line segments were commensurable, it may beassumed that they were looking for such a measure. It can however be easilyshown that if such a line segment exists, then it is also the common measure ofthe pentagon A1B1C1D1E1, whose vertices are formed by the intersections of thediagonals of the original pentagon. In the same way, we could form smaller andsmaller pentagons of the same measure as the original pentagon. This, however,is not possible, and so the assumption about the commensurability of the side andthe diagonal is a wrong one and we have to accept that the opposite is true, thatthese two line segments are not commensurable.

With irrational numbers, three famous problems that the Greeks have been


trying to solve unsuccessfully are also connected with irrational numbers. The�rst of these problems is the duplication of the cube. The following legend isassociated with the problem: an epidemic of the plaque outburst on the isle ofDelos, and as it would not stop, the inhabitants of the isle sent messengers tothe Delf, asking them for advice. The response was easy�it was necessary toduplicate the pedestal of the statue of the god Apollonius, which was a cube madeof gold. There was also the condition that the new altar also has the shape of acube and it must be said that the inhabitants of the isle of Delos could not solvethis problem. The gods, however, probably appreciated their earnest attempts,because the plaque epidemic passed in the end.

Another of these problems is squaring the circle, in other words, constructinga square with the same area as the given circle. While squaring the rectangleis an easy task, as it su�ces to to use the theorem Euclid used in his proof ofthe theorem of Pythagoras v2 = ca · cb and construct a right-angle triangle withthe hypotenuse of the length ca + cb and the height of this triangle is the sideof the square sought, a similar construction for a circle could not be found. Incontemporary �ction and journalism, squaring the circle stands for a problem thatcan only be solved with great di�culties.

The third of these problems is the trisection of the angle, in other words theprocedure of dividing any angle into three equal parts using the ruler and thecompass. Bisection, i.e. dividing the angle into two equal parts, is so easy thatelementary school pupils should master this construction, but nobody succeeded individing the angle into three equal parts. Sometimes, the problem of recti�cationof the circle, i.e. �nding the line segment of the same length as the length of acircle, and the construction of regular polygons, are added to the three problems.All these tasks had to be solved by the so-called Euclidean construction, i.e. withthe ruler and the compass only, which is through a construction of a �nite numberof lines and circles. Today, we know that these problems cannot be solved, becausethe number π is transcendental, or in other words, it is an irrational number whichis not the root of any algebraic equation.

2.3 The Elements

Nowadays, we do not know much about the life of Euclid, but his work Stoichea,in English The Elements, granted him immortality. It is said that it is the secondmost often printed book in the history of mankind, and as the �rst position is takenby the Bible, The Elements have no serious competitor in the category of scienti�cbooks. In this work, Euclid concentrated all the mathematical knowledge of Greeceof his time. It consists of 13 books, whose list and summaries may be found e.g.in [Eu]. There is a certain paradox in that The Elements are the most often readmathematical book in ever, they may also be labelled as the least readable one.The Elements consist namely only of de�nitions, axioms, theorems, lemmas, whileeach theorem is proved here. We will not �nd any examples in the book, neithermotivating the theorem, nor explanatory ones, no connecting commentary; theseare due to later mathematicians and allow better understanding of the text by the

2.3. THE ELEMENTS 17

reader.As the parts that are devoted to geometry, including the famous �ve postulates

of Euclid (these will be introduced in the chapter on the 19th century), are citedquite often, let us rather focus on the arithmetical part of The Elements, i.e. atBooks VII, VIII, and IX. At the beginning of Book VII, Euclid gives twenty-twode�nitions and he begins by de�ning the number one and the (natural) number(D7/1 and D7/2). He thus does not consider one to be a number, but a basicbuilding stone of a number. We can deduce this from the fact that Euclid repre-sents numbers by line segments; one thus represents the basic (unit) line segmentand e.g. number �ve is represented by �ve equal line segments adjacent to eachother. This notion of a number causes the clumsiness of some of the proofs, theattempts to regard one as an ordinary number were not successful.1 The readerswho feel that they have already read about that in this book are not mistaken,since Euclid took this part from the Pythagoreans.

We will now allow ourselves a small detour and cite a few phrases from thetextbook [Sm1], published some 150 years ago. The number (numerus, die Zahl)is several identical objects. One such object coming in that number is called aunit. In the number 7 kreutzers, the unit is represented by the kreutzer, in thenumber ten horses, the unit is a horse, and in the number �ve hundred, the unitis a hundred. This quotation shows that the Priest �imerka wrote his textbookalong Euclidean lines and additionally, the example facilitates our understandingof the thinking of Greek mathematicians.

Let us, however, return to The Elements: further de�nitions concern commonnotions like even and odd numbers, prime and composite numbers, the smallestcommon multiple and the greatest common divisor, ratio, and others. The last, i.e.twenty-second, de�nition then introduces the perfect number, which is a numberthat is equal to the sum of all its parts (read: divisors smaller than the numberitself). The book then continues with thirty-nine theorems concerning the divisi-bility of natural numbers. Here, we will also �nd the famous Euclidean algorithm,although not in the form known to us today. Theorem 1/7: Two di�erent numbersare given and if the smaller one is deducted from the greater and if the numberthat remains is never a divisor of the preceding one, and if we go on deductinguntil only a unit remains, then the original two numbers are mutually prime.

If, for example, a1 = 19 and a2 = 7, then according to Euclid, we need tofollow this procedure: 19-7=12, 12-7=5, 7-5=2, 5-2=3, 3-2=1. It is obvious howto shorten this algorithm and obtain its current version and the proof is thenobvious, because it su�ces to use the fact that from the condition b|a1 ∧ b|a2 itfollows that b|(a1−a2). We will continue this way until we reach the contradictionthat b|1. It is interesting to note that although this theorem is generally valid,Euclid's proof is based on only three deductions. Other general statements are,however, proved in the same way.

Book VIII contains 27 theorems concerning continual proportions or, if youlike, �nite geometric progressions. Let us give theorem 22/8 as an example: Ifthree numbers are continually proportional and the �rst one is a square, then the

1E. g. the Stoic philosopher Chrysippos (280-207 B.C.).


third one is a square as well. The proof of this theorem is easy, because it holdsthat a2 = ka1, a3 = ka2 = kk2a1.

Book IX is devoted to the theory of parity and prime numbers. In it, wewill �nd the theorems that have already been mentioned, concerning parity ofthe sum or product of numbers. In Theorem 22/9, for example, the following isstated: If we add a certain number of odd numbers and that certain number isan even number, then the sum is also an even number. The proof is presented inthe following form: we take four odd numbers, subtract one from each of them,obtaining four even numbers. Their sum is an even number, while adding fourones to it does not change its parity.

In theorem 20/9, the following is stated: There are more prime numbers thanany given number of primes. A proof by contradiction is given; Euclid takes threeprimes, �nds their smallest common multiple and adds one to it. A number thusoriginating is either a prime (but that would be a fourth one), or a compositenumber. However, it cannot be divisible by any of the previous one, so we haveforgotten about one. The Greeks did not have (our set-theoretical) notion of actualin�nity, they only understood it as potential in�nity, namely in the sense that theycan increase the given number. For that matter, in the geometrical part of TheElements, the straight line is not de�ned as in�nite, but as a line that can beextended without limits beyond either of the two endpoints.

Theorem 35/9 is in fact the formula for the sum of the �rst n members ofgeometric sequence: Let any collection of numbers be given, all continually pro-portional. We subtract the �rst number from the second one and from the lastone. Then the surplus of of the second in ratio to the �rst one is the same as thesurplus of the last one in ratio to the sum of all the preceding ones. This ratherclumsy wording says, in modern notation, that a2−a1

a1= an+1−a1

a1+a2+···+an. Deducing

the well-known formula for the sum of �rst n members of a geometric sequence isthen very easy.

The last theorem of this book (36/9) runs as follows: Let us have any amountof numbers beginning with one which are continually proportional with the quotienttwo and let us add these numbers until their sum is a prime. If we multiply thissum by the last number [added], the product will be a perfect number. Here, Euclidstates a su�cient condition for a number of the form (2n−1).2n−1 to be perfect. Itcan be proved that this is also a necessary condition. Numbers of the form 22 − 1are called Mersenne numbers and nowadays, roughly 40 of them are known thatare also prime numbers and searching for Mersenne primes is nowadays a big hitfor amateur mathematicians.

To conclude this chapter, I dare to present a brief contemplation concerningEuclid and the contemporary world. Euclid's Elements are based on the opinionsof two well-known philosophers of the antiquity. The �rst one of them is Plato 2

and his idealist philosophy, which di�erentiates between two worlds, the real one,in which we live, and the transcendental world of ideas. In our world, there arepeople, animals, plants, and things that we get to know through our senses. Theseoriginate, change over time, and �nally perish. A trembling pinscher is a dog, justlike a masti� is. If we disconnect a couple of carriages from a train, what will

2Plato (427/8-347 BC), Greek idealist philosopher, who founded the school called Academia.

2.4. PRACTICAL MATHEMATICS 19

depart from the station will still be a train. Transcendental world, however, canonly be recognized by reason, is perfect and the only real one, and it is governedthe idea of the Good. In this world, only one dog and only one train exist. Theidea of the dog and the one of the train is not born and does not die, it does notre�ect anything else in itself and it does not change into anything else either. Theidea of the dog is naturally contained in all dogs that run around the earth, justlike the idea of the train is contained in all trains that have ever traversed, aretraversing, or will traverse the railroads. The idea itself, however, is not in anyway in�uenced by the concrete realisation.

This is the way things are also in The Elements. De�nition 1 states: A pointis that which has no parts. De�nition 2 tells us that A line is a length without awidth. However, even if we sharpen a pencil with the most precise pencil sharpenerequipped with the most modern technology, even then the points drawn with thispencil will be speckles and the line drawn magni�ed appropriately will resemblea motorway. As Vlasta Burian would say, geometry has abstracted itself fromthe reality. We solve geometrical through reason and then we more or less exactlydemonstrate them on paper. It is thus no coincidence that the following inscriptionwas there above the entrance to Plato's academy: Do not enter, if you do not knowgeometry.

The second one is Aristotle from Stagira,3 the founder of logic. According tohim, it is necessary to devote ample attention to the de�nitions of notions, becausepeople communicate through notions. The road of recognizing the principles leadsfrom from the individual to the general and it is called induction, these principlesthen have to be justi�ed through judgement or deduction. If it is shown thatthe new principle is a consequence of already con�rmed principles, then we haveperformed the proof. Not every statement can be proven, and such statementsare called axioms. The Elements are constructed logically in this sense and othermathematical theories are also based on this principle

2.4 Practical Mathematics

From the previous lines, it might seem that the Greek scientists only solved the-oretical tasks in the quiet environment of their studies. The Greeks, however,were also adept sailors, which can be seen from their mythology (Argonauts, theOdyssey, etc.), and also from the fact that they managed to colonize major partof the Mediterranean. There were seven wonders of the old world, as Philo said,and of those, �ve were on the territory inhabited by the Greeks and at least three(Temple of Artemis in Ephesus, Mausoleum in Halicarnassus, and the Lighthouseon the Pharos Island) were especially wonders of architecture. These facts arealready an evidence of the fact that they could use mathematical knowledge inpractice.

If we ask any passer-by about Archimedes, they will probably think about theobject partially immersed in a �uid and will conclude that he was a physicist. If

3Aristotle (384-322 BC), leading personality of the Lyceum, polyhistor with great sense forthe methodology of scienti�c examination.


we take into account current division of the sciences, we can of course agree withthis statement, but we should not forget that Archimedes was also an excellentengineer and that from the work that has been preserved until today it is clearthat we can also classify him as a mathematician. In any case, we can say thatArchimedes was the most signi�cant scientist of the antiquity. Pupils meet hisname already at elementary school, so we will deal with some of his discoveries inmore detail.

While the calculation of the perimeter or area of regular polygons does notpresent any di�culties, the determination of the formulas for the perimeter of acircle or the area of a circle is by far not easy and it requires much inventivenesson the part of the teacher to explain the formulas so that the pupils understandit, because the pupils have not yet acquired the necessary mathematical tools. Itwould nevertheless be worthwhile to return to this issue once after introducinggoniometric functions and attempt to derive the formulas in a manner similar tothe one used by Archimedes many centuries ago.

The method of Archimedes is based on the exhaustive principle of Eudoxos,which can be formulated in the following way: Let l be the length of a circle, ln isthe length of a regular polygon inscribed into this circle. Then for each k > 0 thereexists n such that l − ln < k. In a similar way, we can formulate this principlealso for the length of a circle and the regular n-gon circumscribed to the circleand in a similar way, we can proceed also for the area of polygons inscribed orcircumscribed to a disk.

Let us thus have a circle with diameter d and let us ask by what number weshould multiply the diameter to get the length. If we denote that number by π,then we will obtain l = πd. For the circumference of a regular polygon inscribedto the circle we can easily derive the formula

ln = nd sin360◦

2n(2.1)

.By comparing both formulas, we obtain lower bound for the number π, namely

π > n sin 360◦

2n . In a similar manner, we can obtain the upper bound because thecircumference of a regular polygon circumscribed to the circle is

Ln = nd tg360◦

2n(2.2)

and it therefore holds that π < n tg 360◦

2n . Through connecting these inequalities,we obtain the bounds for number π, given by the following inequality:

n sin360◦

2n< π < n tg

360◦

2n. (2.3)

From these considerations, it is obvious that we can never obtain the exactvalue of the number π this way. For practical calculations, however, it su�ces toknow only an estimated value of this constant. It therefore su�ces to determinethe accuracy k in advance and to calculate the values of both estimates until weget Ln− ln < k. If we are not calculating these values with the help of a computer,

2.4. PRACTICAL MATHEMATICS 21

but only with a calculator, it is su�cient to start with a certain n and double thenumber of sides of the polygon until the required inequality holds.

We can proceed in a similar way when determining the constant by which weneed to multiply the square of the diameter (radius) of a disk to obtain its area.For practical reasons, we will seek a number by which we need to multiply thesquare of the radius and we will again denote this number by π. The followingformula can easily be derived for the area of an n-sided polygon inscribed into acircle:

pn = nr2 sin360◦

2ncos

360◦

2n=

1

2nr2 sin

360◦

n(2.4)

.The area of a circumscribed n-sided polygon is then given by the formula

Pn = nr2 tg360◦

2n(2.5)

. Thus, we have the following bounds for number π:

n sin360◦

n< π < n tg

360◦

2n(2.6)

and we proceed as before to determine number π.Archimedes started from a regular hexagon and he stopped his calculations

when he reached the regular 96-gon. It is usually stated that the used the valueof π = 22

7 , which corresponds absolutely with the value π .= 3.14, which is a value

su�ciently exact for the practical purposes in technical calculations.Archimedes also worked on the quadrature of the parabola, i.e. calculating the

area of a rectangular triangle whose hypotenuse is substituted for by the parabolay = x2. In one of the ways, he used his knowledge of mechanics in an ingeniousway. To illustrate his method, we will use current symbolic language to facilitatethe understanding. Let us thus have a lever which we identify with the x axis andwhose centre of rotation will be the origin of the coordinate system. On the lefthand side, we will place an isosceles triangle whose sides will be y = 1

2x, y = − 12x

and x = −1. On the right hand side we then put the parabola y = x2, wherex ∈ [0; 1]. If we take any x0 from this interval, then also the transversal line ofthe triangle has this length and the momentum of the force will be M = x0 ·x0%g.To achieve balance, we need to take also the transversal of the triangle on theleft-hand side and place its end to the point x = 1. The whole triangle on theleft-hand side can thus be equilibrated in such a way that we concentrate all thetransversals of the triangle on the right-hand side in the point x = 1. As thedensity and gravity compensate each other and the triangle may be substitutedfor by a mass point placed in its centroid, we obtain the equation

P (x) · 1 =x2

2· 2

3x⇒ P (x) =

1

3x2.

This, however, is not the only way in which squaring of the parabola can beachieved. Archimedes also used the exhaustive method and �lled out the areaunder the parabola with triangles, see [Zn]. Apart from squaring the parabola, he


also derived the formula for calculating the volume and surface of a cylinder andof a sphere. He also dealt with the issue of a sphere inscribed into an equilateralcylinder and found out that the rations of the volumes and surfaces of these solidsare 2:3. This curiosity can also be easily derived when dealing with the topicof volumes and areas of solids already in elementary schools. When Cicero wasa quaestor in Sicily, he found Archimedes' grave and discovered that a sphereinscribed into a cylinder was carved on the tomb, thus probably revealing howmuch the genius treasured his discovery. Evil tongues say that Cicero's discoveryis one of the most important contributions of the Romans to the development ofmathematics.

To end this part, let me add a few philosophical considerations. While Euclid'sapproach can be described as static and purely intellectual, Archimedes' methodscan be described as dynamic and engineering-like (practical). It would probably berather courageous to maintain that Archimedes used in�nitesimal calculus in hisconsiderations, but we can indeed �nd the basic ideas of this branch of mathematicsin his work. This Syracusan was able to introduce mathematics into practice, aswe have shown brie�y, and perhaps because of that he was greatly admired bythe author of the book [Be]. Here we can agree with the author of that book,but we do not agree with his despect of those scientists who belonged to the �rstkind. We think that the Euclidean constructions have their place in the teachingof mathematics; although only few people will have to, in their lives, construct atriangle given by a side and the median and the altitude to that side. Preciselythese constructions, as well as proofs and similar tasks, play an indispensable rolein the development of logical thinking.

Chapter 3

Mathematics in the Middle

Ages

In 476, the last emperor of the Western Roman Empire, Romulus, was dethroned.It is a paradox of history that although his name was the same as the nameof the mythological founder of Rome, the historians gave him the appellationAugustulus (mini-emperor). This date is often taken to be the breaking pointbetween the Antiquity and the Middle Ages, although the eastern half of theRoman Empire did not break down under the raids of the barbarians and lastedfor another thousand years. On the debris of the Western Roman Empire, empiresof various Germanic tribes rose and fell. The most signi�cant and also the moststable state was founded by the Franks, whose empire became the foundation oftwo big current European states - France and Germany. The most signi�cant andthe best known ruler of the Frank empire was Charles the Great, who understoodthe importance of education and on whose court Alcuin, the learned monk, lived.Charles's empire did not last long after its founder had died, but current leadingEuropean powers, France and Germany, were founded on its basis.

3.1 Alcuin and the others

If you take a book dealing with recreational mathematics, you will most probably�nd the following puzzle in it: A man needed to transport across the river a wolf,a goat, and cabbage and he could not �nd another boat then such which couldtransport only two of them. However, he had been told that he should transportthem all unharmed. If you know, advise how he could transport them unharmed.

This problem is already around twelve hundred years old and I think that itwill entertain even today and those who do not know it have to think for a whilebefore they �nd the solution. It was �rst stated in the book Propositiones adacuentos iuvenes, which can be translated as "Problems to sharpen the young",whose author is thought to be Alcuin himself. Apart from this problem, we canalso �nd three other problems of this kind; Alcuin is namely thought to be the

23

24 CHAPTER 3. MATHEMATICS IN THE MIDDLE AGES

inventor of this kind of problems by historians of mathematics. Let us state onemore of his problems, which is not so well known: There were three men, eachhaving an unmarried sister, who needed to cross a river. Each man was desirousof his friend's sister. Coming to the river, they found only a small boat in whichonly two persons could cross at a time. How did they cross the river, so that noneof the sisters were de�led by the men? From the solution given by Alcuin himself,it is clear that a sister would be de�led if she found herself alone with anotherman without her brother present. The solution to this problem implies that therequirements on the morals of women in the times of Alcuin were di�erent fromthose of today.

A couple of years ago, the author came across a problem that is analogous to theprevious one and which should allegedly be used in Japan as a test of intelligenceduring job interviews. The time allowed for solving the problem is half an hour, andas the author still works in Brno, it is clear that he has not solved it. However,there is a solution to the problem: A father with two sons, a mother with twodaughters, and a policeman with a captive. They have a boat for two people, whichmay only be driven by an adult. The sons cannot be left alone in the presence ofthe mother and the daughters cannot be left alone in the presence of the father.The captive cannot be left alone with any member of the family, but interestingly,he will not run away while not accompanied by the policeman. Frank youngsters,however, did not sharpen their minds only through problems about carrying thingsover water, but also through others. A well-known monologue of Vlasta Burianabout confusion among relatives (already Jaroslav Mo²na entertained the publicby similar ones) is modelled on a problem from Alcuin's book, namely the followingone: Two men, a father and a son, marry two women, a widow and her daughter,while the father marries the daughter and the son the widow. Tell me, I ask, whatthe relationship will be between the two sons born into these marriages.

Prevailing part of the collection is, however, devoted to mathematical tasks.In it, we may �nd problems leading to equations with one unknown, geometricalproblems, problems involving conversion of units and also tasks involving sequencesas well as problems leading to Diophantine equations and combinatorial problems.Most of these tasks are similar to those we can �nd in today's textbooks, onlythe way of posing the problems is much richer than today's strict tasks. Whenwe realise that in the 8th century, there was no algebraic notation as we know ittoday at hand, we have to admit that solving a word problem by the pupils of thattime was a valuable achievement. It is, however, also quite possible that if a timemachine transferred students of those times into the 21st century, they would notunderstand our approach either and might wonder why we solve such easy tasksin such a complicated manner.

Alcuin was not the only author whose collection of problems survived untilthese days. Four medieval problems are ascribed to Venerable Bede, of whichthe fourth one is interesting because it speaks of negative numbers. The collec-tion ascribed to the Greek poet Metrodorus probably originated in 5th centuryByzantium, as the problems were published in the Greek Anthology, written byMetrodorus. It is, however, also possible that the problems themselves were writ-ten by a di�erent author and Metrodorus only versi�ed them. A collection of six

3.1. ALCUIN AND THE OTHERS 25

problems by Abu Kamil, called Book of Rare Things in the Art of Calculation,survived from the 10th century. The solutions to these problems are always foundthrough a system of Diophantine equations and their formulation is rather stereo-typic: a person has 100 drachmas and their task is to buy several birds for variousprices. For more information, the reader is advised to consult [Ma1]. We presenthere a selection of tasks from this book as an advertisement.

Today's pupils would solve the following problem using a linear equation withone unknown: A boy greeted his father: Be greeted, father, he said. The fatheranswered, be well, son. May you live as long as you have lived up to now and mayyou triplicate this double quantity of the years and add one of my years and youwill live for a hundred years. Let answer he who knows how old the boy then was.Apart from mathematics, we should also appreciate how nicely the father and theson behaved to each other.

Today's tasks on the conversion of units are usually monotonously formulatedthrough a brief command to convert decimetres to centimetres and a few taskswith concrete numbers follow. Alcuin, however, formulates these tasks with graceand the task chosen reminds one rather of the tales by Fontaine or Krylov: Asnail was invited by a swallow to lunch a league away. However, it could not walkfurther than one inch per day. How many days did it take for the snail to walkto that lunch? The conditional in this task is appropriate, because according toAlcuin, snail would walk to the lunch in 90,000 days.

Problems about common work are usually set in a manner accurately describedby Stephen Leacock in Literary Lapses. Superman A would almost completethe whole task by himself, B is a normal worker and C is so clumsy that onewonders how the two could have him on their team. Not so for Metrodorus:(Dear) manufacturer of �red bricks, I very much wish to �nish my house. Todaythe sky is cloudless and I do not need many more �red bricks; together there areonly three hundred that are missing. That many you have made yourself in oneday and two hundred were made in a day by your son and as many as that and�fty more were supplied by your son-in-law. How many hours does it then lastwhen the three of you set out to work together? Apart from the poetic value, wewill also appreciate the realism of the problem. The son and the son-in-law helptheir father signi�cantly, while it is understandable that the master is a bettermanufacturer than the journeymen.

Relations among the various teaching subjects can be strengthened by problemsin which �gures from Greek mythology are mentioned. Let us state one of them toend this section: Once Aphrodite asked Eros, who came to her distressed. "Whatmakes you so sad, my child?" He then answered: "I was coming back from Helicon,with a load of apples, but Muses have robbed me of those apples and then ran away.Clio took a �fth of them, Euterpe a twelfth of all the apples; then Thalia, the nobleone, took an eighth, Melpomene a twentieth part, Terpsichore a quarter, and Eratotook a seventh part as her share. Polyhymnia stole thirty apples, Urania then ahundred and twenty, Calliope stole away with a heavy load of three hundred apples.And then I came home to you - look - almost empty-handed; the goddesses onlyleft me with �fty-�ve apples. As the Muses have been and still are an inspirationfor artists as well as scientists, we will forgive them this lapsus.


3.2 Fibonacci, or problems not only of breeding

Somebody placed a pair of rabbits at a certain place surrounded by walls on allsides to learn how many rabbits would be born during a year, if by rabbits it is sothat a pair of rabbits gives birth to a new pair of rabbits every month and if rabbitsbegin to give birth when they are two months old. As the �rst pair gives birth tonew rabbits in the �rst month, multiply by two, and in this month you will havetwo pairs, from these pairs, one, namely the �rst one, will give birth also in thecoming month, so in the second month, you will have three pairs. From the threepairs, two will give birth in the third month, so you will have two more pairs inthe third month and the number of rabbits in this month will reach �ve...

This problem can be found in the book Liber abaci, written by Leonardo Pisano,also known as Fibonacci. The author describes the situation in the individualmonths, until he �nally comes to the conclusion that there will be 377 pairs ofrabbits in that place at the end of the year. The solution to the problem endswith a note that we can count rabbits in this way up to in�nity. If we disregard theslightly idealized conditions, as faithfulness of partners and sound health on oneside and the unbelievably low and surprisingly regular reproduction, we have justbeen shown an interesting sequence. This sequence is nowadays called Fibonaccisequence and can be de�ned by a recurrent formula Fk = Fk−1 + Fk−2, F0 = 0,F1 = 1.1 The numbers in this sequence are called Fibonacci numbers and apartfrom mathematics, we can �nd them a.o. in Dan Brown's book Da Vinci Code.

Nowadays, we can meet Fibonacci numbers in various parts of mathematics,and we will mention an interesting connection here. In statement 11 of his BookII, Euclid solved the following task: Divide a given line segment in such a waythat the area of the rectangle whose one side is the original line segment and oneof the parts is the same as the area of a square whose side is the other part.

If x denotes the side of the square, then it must hold that

a(a− x) = x2,

which leads to the quadratic equation x2 + ax − a2 and its solutions are x1,2 =a(−1±

√5)

2 . Only the positive root ful�ls the conditions of the problem. Let usdenote

Φ =a

x=

x

a− x,

then after substituting for x, we obtain Φ = 1+√5

2

.= 1, 618034. The problem of

dividing a line segment into such parts is nowadays called the golden ratio andthe number Φ the golden number. This number is also equal to the limn→∞

Fn+1

Fn.

Further interesting details about this number can be found in [Ja], including morereferences to the sources.

Fibonacci was the most signi�cant central European mathematicians and thesequence named after him is by far not his only contribution to mathematics.Apart from the already mentioned Liber abaci, he is also the author of four othermathematical treatises, which we will introduce by the translated titles: Practice of

1The name was �rst used by the French mathematician E. Lucas (1842-1891).

3.2. FIBONACCI, OR PROBLEMS NOT ONLY OF BREEDING 27

Geometry, Blossom, Letter of the undersigned Leonardo to Master Theodorus, theImperial Philosopher, and Book of Squares. Although his works are not alwaysoriginal and although he often takes over the mathematical knowledge of othermathematicians, whose writings he probably encountered during his journeys, hiswork deserves attention. We cannot deal with his writings in detail here, theinterested reader may consult [Be2], therefore we will only brie�y introduce heresome of his results.

Fibonacci was one of the �rst mathematicians to use Hindi-Arabic numeralsand the decimal position system. Part of his work is devoted to the writing ofnumbers, numerical operations including �nding the square root and conversionof units. In Liber abaci, we will �nd also problems solved by equations, linear aswell as quadratic and cubic ones. As our contemporary notation was not used inthose times and negative numbers were not calculated with, quadratic equationshave to be divided into several classes, so that there are no negative coe�cients.With quadratic equations, we can basically reconstruct the solution and concludethat we basically can use our formulas, but this does not hold for cubic equations,although the solution found by Fibonacci is very precise.

Leonardo Pisano worked also with sequences; apart from the already mentionedone which is named after him, we can �nd problems leading to the sum of the �rstn members of the arithmetic sequence, problems leading to the sum of squares,and the ever-green of recreational mathematics, namely the Ancient Egyptian taskabout cats. Since many years have passed since this problem was �rst posed, in hisversion, the cats became old women. In his works, we can �nd also problems solvedthrough Diophantine equation. Apart from the problems about birds found at AbuKamil, we can also �nd here more complex tasks, resembling rather Aritmetica byDiophantus. We can present this problem as an example: Find a square numberwhich, increased by 5 and decreased by 5, gives a square number.

When de�ning basic notions in geometry, he followed Euclid. He also devotedtime to "measuring �gures", which is a notion under which we must see especiallythe instructions for calculating the area. He provided general formula for calculat-ing the perimeter and the area of a circle; in his version, π = 22

7 . It is interestingthat he reproduces also Archimedes' method for calculating the value of the πthrough circumscribing and inscribing a regular 96-gon. Fibonacci also providesinstructions for the calculation of the volume of solids, especially the cone and thesphere. He also pays attention to the calculations of the lengths of regular pen-tagon and decagon and the golden ration. He was able to prove that the mediansof a triangle meet in a single point and that this point divides each of the mediansin the ratio 2:1.2

To end this section, let us mention two curiosities. In the works of Fibonacci,there are signs of using mathematical notation and of attempts to solve problemsin a general way. We will present the following problem as an example: Fivehorses consume six weighing units of oats in nine days. In how many days willten horses consume sixteen weighing units of oats? Fibonacci says that a horsesconsume b weighing units of oats in c days, and d horses e weighing units of oatsin f days. Then it holds that abc = def and from that equation, we can determine

2This fact was known already to Archimedes, but his proof has not been preserved.


the demanded value. Some problems are about money of several people; theseproblems lead to a system of linear equations with one negative solution. Fibonaccistates that such a problem is solvable only when the given person is in debt (inother words, the solution is a negative number).

3.3 The Arabs

We have already encountered the Arabs in the part devoted to Egyptian mathe-matics and the note on their contribution to science was very negative. However,we have suggested that it was an exceptional case and now the right moment comesfor us to introduce the reader to the Arabic contribution to the treasure box ofworld mathematics. When Mohammed �ed from Mecca to Medina in 622, notmany people probably suspected that he would victoriously return in eight hun-dred years later and that his descendants would conquer large territory on whichthe new religion - Islam - would be spread.

Over a few centuries, the Arabs conquered a territory that reached from Spainin the west and over North Africa and the Middle East up to the river Indusin the east. This empire was rather heterogeneous and it gradually broke upinto several independent states and the empire gradually declined and lost thepositions it had conquered. Similarly to the situation in the Roman Empire, nativepeoples continued to live on the territory conquered by the Arabs, and thereforethe mathematics that was developed there is sometimes called Islamic mathematicsrather than Arabic mathematics. However, this term is not exact either, as otherreligions, especially Christianity and Judaism, were tolerated on the territory. Tomake things simple, we will stick to the name that has traditionally been used inthis country, namely Arabic mathematics.

Just like the Ptolemaic dynasty in Egypt, the caliphs of Baghdad were alsoaware of the importance of science and education and they supported scienti�cresearch in all possible ways. During the reign of Harun al-Rashid (786-809), alarge library was established, which was continually supplied with new books. Hissuccessor, al-Ma'mun, then founded something like Mouseion. He called this insti-tution House of Wisdom and concentrated top thinkers of his time in it. Similarinstitutions also originated in other places. In a simpli�ed form, we can say thatArabic mathematicians acquainted themselves with the results of their colleaguesin other cultures (Greece, Babylonia, India, etc.) and that they further developedthe knowledge they thus learnt. We should be grateful to the Arabic thinkersthat they translated Euclid's Elements, Diophantus's Arithmetics, the works ofArchimedes, Apollonius, and other Greek scientists. Some of their �ndings wouldprobably have been lost without the translations. Ptolemy's astronomical workSyntaxis megalé, for example, is much better known under its Arabic title Al-magest. The translations were accompanied with thoughtful commentaries, whichmade these works much more comprehensible. The Arabs also freed themselvesfrom geometric ideas in arithmetic.

One of the parts of school mathematics is algebra. This title comes from theArabic and was taken from the title of the most famous Arabic mathematician,

3.3. THE ARABS 29

al-Khwarizmi, whose title is A short book about the calculations through al-jabr andal-muquabala. Apart from practical problems, (commercial contracts, testaments),he also treats solving linear and quadratic equations with integer coe�cients in thisbook. Al-Khwarizmi does not use symbolic notation, and thus he has to describeall operations in words. He does not include negative numbers in his consider-ations, and therefore he has to consider several types of equations with positivecoe�cients. In the same way, he does not include negative solutions. When solvingthe equations, he uses two types of operations: al.-jabr, which basically consists inadding the same member to both sides of the equation, and al-muquabala, whichconsists in adding up the members of the same degree. Quadratic equations, how-ever, were not the top achievement of Arabic mathematics. Omar Khayyam wrotea book on the classi�cation and solving of equations of the third and fourth degreesand al-Kashi could solve some equations of fourth degree. In Arabic mathematics,attention was paid also to trigonometry, a discipline in which they reached con-siderable achievements. They drew upon the work of Greek (Ptolemy) as well asIndian mathematicians. The idea to measure the angle with the aid of the chordoriginated in Greece, and in Indian mathematicians improved it through the useof half of the chord, namely the sine function. The Arabs took over this knowledgeand developed them. They introduced new functions, a shadow (cotangents) anda reversed shadow (tangents) and they also used the functions now half-forgotten,secant and cosecants. In order to simplify the calculations, tables of trigonometricfunctions were calculated. It is interesting that they only used this knowledgeonly for solving problems in astronomy. They solved problems in a plane similarlyas the Greeks, namely by dividing the general triangle into two right-angle trian-gle. They proved the sine theorem and formulated the cosine theorem, withoutattributing it any signi�cance.

The Arabs left one more signi�cant trace in mathematics, one that we meet allthe time, namely Arabic numerals. Already al-Khwarizmi realised how importantit is that we can describe any quantity using only a set of nine symbols. There areactual ten symbols, since al-Khwarizmi uses decimal positional system of notationand for the sake of unambiguity, it is necessary to have a symbol for the free space.In his version, it was a circle, i.e. our later symbol for null. Decimal positionalsystem, however, is not an Arabic invention, but an Indian one. The Indians havebeen gradually adopting this system since the 7th century. Other nations graduallytook it over from them. As in India, in the Arabic countries, this system was onlygradually, but steadily, adopted. It reached Europe through Spain, at that timegoverned by the Arabs (the Moresques). The oldest manuscript using Arabic (weshould rather call them Indian) numerals is from 976 and it was found in northernSpain close to Logro?o. Decimal positional system gradually substituted mainlythe impractical way of writing down numerals in the Roman way. However, as faras units of length and weight, which are mainly not based on the decimal system,are concerned, the tradition has largely been preserved until today.3 At the endof this section, let us add a small linguistic note. Latin name of al-Khwarizmiwas mostly Algoritmus and this name became the name of the new arithmetic.

3Even Good Soldier �vejk maintained that there should not be ten, but twelve �ags inKonopi²t¥, since in when in dozens, things are always cheaper.


Arabic name for the empty space, null, was as syfr. The word zero as well as theword cipher (now used in the sense of digit in Czech) originated from this word.Null comes from the Latin word nullus (none) and it permeated into the commonspeech of mathematicians in the 15th century.

3.4 Mathematics in the Czech lands

As was already mentioned, the �rst evidence of mathematical thinking, a notchedrib bone, was found in Dolní V¥stonice in southern Moravia. Let us have a lookat how our predecessors developed mathematics during the middle ages. Thiswill be di�cult, as hardly any purely mathematical works from this period, andespecially from the early Middle Ages, have been preserved. We therefore haveto rely on material or, alternatively, indirect evidence. The �rst state on ourterritory was Great Moravia. All four rulers of Great Moravia were Christiansand this religion was gradually adopted on the whole territory. In the towns ofGreat Moravia, churches were built and, as archaeological �ndings show, these werebuilt in a certain module, and therefore it is certain that the old Moravians hadsound technical knowledge. Old Moravians also had lively commercial relationswith neighbouring countries, and as the systems of measuring and weighing unitswere then not at all uniform, the merchants of that time had to be well versedin unit conversions. The most important Christian holiday is Easter, but thisis a movable feast. Given the limited communication possibilities, this feast wasannounced by the priest in each church, so they at least had to know the algorithmfor the calculation of the dates.

Towards the end of 9th century, practically with the fall of the Great MoravianEmpire, the house of Premyslids was on the rise and they soon reigned the wholeof Bohemia and thanks to B°etitslav, also the territory of Moravia. Mathematicalknowledge of this period can be described in a similar way to that during GreatMoravia. The development of feudalism coerced further developments, especiallyin geometry. It was namely necessary to determine exactly the area of �eldsand forests, and by the same token, it was necessary to divide exactly the landwhen founding villages and towns. Surveying thus became an important branchof human activity and its level was very good.4

We know from the legends about Saint Wenceslas that the future prince andsaint Wenceslas attended the school at the church of St. Peter and Paul in Bude£.This school educated the sons of the foremost noblemen, especially in order toprepare them for the duty of the priest. Similar schools were founded also at otherchurches. The school at the chapter of St. Vitus, where studium generale minorwas founded, was the most famous one. The knowledge graduates from theseschools obtained were on a par with the standard in Central Europe at that time.They learned to add and subtract, halve and duplate, multiply and sometimesalso divide. Teaching to calculate with fractions and algorithms was less central.

4The most signi�cant proof of the level of Czech surveying is New Town in Prague, foundedby Charles IV. in 1348. The king and his surveyors outran their time by several centuries. CattleMarket (Charles Square) became the largest square in Europe and it kept this primacy untilthese days.

3.4. MATHEMATICS IN THE CZECH LANDS 31

The range of natural numbers was approximately one million. In geometry, basicnotions like point, line, surface etc. were used. The procedures to calculate theperimeter and area of basic �gures in the plane as well as for the calculation ofthe volume of some solids.

The founding of the university in Prague in 1348 was a signi�cant impulse forthe development of education (not only) in Bohemia. Charles IV thus succeeded inful�lling what his grandfather, Wenceslas II, tried to accomplish in vain, namely tofound universitas magistrorum at studentium. Prague, nowadays Charles, univer-sity was the �rst one of its kind north of the Alps and students from other centralEuropean countries, not just from the Lands of the Bohemian Crown, studiedthere. We will be primarily interested in the faculty of arts whose role was di�er-ent from the faculties of arts today. Philosophical (artistic) faculty was a sort ofa preparatory course for the study at the faculties of medicine, law, and theology,and among other things, mathematics was also taught there. This faculty may alsobe called the faculty of seven liberal arts: the humanities-oriented trivium (gram-matics, rhetoric, and dialectics) and the sciences-oriented quadrivium (arithmetic,geometry, astronomy, and music).

The founding of the university in Prague in 1348 was a signi�cant impulse forthe development of education (not only) in Bohemia. Charles IV thus succeeded inful�lling what his grandfather, Wenceslas II, tried to accomplish in vain, namely tofound universitas magistrorum at studentium. Prague, nowadays Charles, univer-sity was the �rst one of its kind north of the Alps and students from other centralEuropean countries, not just from the Lands of the Bohemian Crown, studiedthere. We will be primarily interested in the faculty of arts whose role was di�er-ent from the faculties of arts today. Philosophical (artistic) faculty was a sort ofa preparatory course for the study at the faculties of medicine, law, and theology,and among other things, mathematics was also taught there. This faculty may alsobe called the faculty of seven liberal arts: the humanities-oriented trivium (gram-matics, rhetoric, and dialectics) and the sciences-oriented quadrivium (arithmetic,geometry, astronomy, and music).

Since the foundation of the university, many teachers took part in the teaching,of whom we will mention two who were probably the most important. The �rstof them is Master Christianus de Prachaticz (?1368-1439), a contemporary anda friend of Jan Hus and also a vicar and a priest at St. Michael church in theOld Town in Prague. His work is mostly not original; he adapted the work of for-eign scholars (Sacrobosco, Johannes vom Erfurt). Algorismus prosaycus magistriChrisitianni, in which he describes basic arithmetical operations, is considered themost important. His Computus chirometralis is devoted to composing the calen-dar, determination of movable feasts, cycles of the Sun, etc. His other works dealwith astronomy and also with medicine.

The name of another important mathematician is known even less, even thoughthe inhabitants of Prague may daily see one of his creations and that this workis also admired by many tourists from the Czech Republic as well as from othercountries. This work is Prague horologue, which consists not only of the towerclock, but also of a planetarium. According to Alois Jirásek and his AncientBohemian Legends, this outstanding piece is the work of the genius named Hanu²


from Rose, who was blinded by Prague councillors so that no other town couldboast with such a beautiful horologue. This legend is romantic, but it is far fromreality. The horologue was constructed by the foremost Bohemian clockmakerand mechanic, Nicolas from Kada¬. He, however, lacked the necessary knowledgeof astronomy, so it is clear that he had to construct the work according to anoutstanding astronomer. This astronomer was nobody else than Jan �indel (1373-?1450). His astronomical works have not been preserved, but they must have beenoutstanding, because they were also valued by world-renowned astronomers of theRudolphine times, Kepler and Brahe. Like Christiannus, he also devoted attentionto medicine.

Palacky considers the Hussite period the most important one in Czech history.He de�nitely had good reasons for this statement, since apart from success in the�eld of politics and the military, the Lands of the Bohemian Crown were alsoknown for their religious tolerance, which was a phenomenon unheard of duringthe middle ages, but this period was not very favourable for the developmentof mathematics and the sciences. It cannot be said that mathematics was notdeveloped at all, but its level remained basically the same until the time when theeccentric emperor Rudolf II moved to Prague. We will describe the developmentsduring that time in the next chapter.

3.5 More names

So far, we have only introduced names and works that can be used in teachingmathematics at elementary schools and high schools. However, the Middle Ages inEurope were not only a period of stagnation of science and apart from the thinkersalready mentioned here, there were others whose contribution to mathematics wasnot insigni�cant. In the following brief account, we will introduce a few othernames. Anitius Manlius Boethius (around 480-524), also known as the last Roman,worked in the court of the king of the Germanic Ostrogoths, Theoderic the Great.He was the author of textbooks for all the disciplines of quadrivium and thanksto him some knowledge of the Greek mathematicians was preserved to these days.His work was namely not original, but consisted in translations of Ancient Greekauthors (Euclid, Nicomachus, etc.). His translation of Arithmetics by Nicomachuswas used as a textbook for nearly a thousand years.

Gerbert of Aurillac was one of the greatest thinkers at the turn of the secondmillennium. The date of his birth is not known; he was from a poor, but free family.He was brought up and educated by the Benedictines from a nearby monastery. Hisinterests were very wide: during his travels, he for example met with the work ofArabic scholars. His mathematical work was not original, but it contributed to thespread of mathematical knowledge in Europe. He contributed to the renaissanceof the abacus, which he had improved. He was probably the �rst European whogot acquainted with Arabic mathematics and to a limited extent used Arabicnumerals. 5 His work Geometria deals with solving some geometrical problems,

5On his abacus, we can �nd symbols similar to the Arabic gobar numerals. In his writtenwork, however, he writes numbers out in words or he uses Roman numerals.

3.5. MORE NAMES 33

but the solution is given without proofs or detailed explanations. Some chapters inthis work are also devoted to surveying. He was also an outstanding astronomer,in Lateran palace in Rome; he allegedly had an observatory constructed. He wasalso a major European politician and an advisor to Emperor Otto III, who aspiredto unify the Christian world under one emperor and pope. In his church career,he reached the highest goal, in 999 he was the �rst Frenchman to be elected popeand he accepted the name Silvester II. After four years in the o�ce, he died andthe circumstances of his death are not clear. He is buried in the basilica of St.John in Lateran. There are many legends about his life, not always favourable forhis role as a priest. As far as the author knows, mathematicians, in opposition toother professions, do not have a patron saint. Gerbert would probably be the bestcandidate for this position, but he was never canonized.

Nicole Oresme (?1323-1382) was the most signi�cant personality of Europeanmathematics in the 14th century. His most important mathematical work is Al-gorismus proportionum. In this work, he introduced the powers with positiverational exponent and stated rules for calculating with such expressions. In his Delatitudinibus formarum, he applied dependent variable (latitudo - width) againstthe independent variable (longitudo - length), which can be changed. It is thussome kind of transition from coordinates on the heaven or earth sphere, whichalready Ancient scholars used, to geometrical coordinates as we know them now.It is probable that this work in�uenced also the founders of analytic geometry.

Johannes M¶ller called RegiomontanusJohannes Müller, lived in the 15th cen-tury. This excellent computer, but also a mechanic and a typographer also par-ticipated on translations and editions of classical mathematical manuscripts. Hismain work is called De triangulis omnimodus libri quinque, which is a systematicintroduction into trigonometry. Regiomontanus could not use our contemporarysymbolic expressions, and therefore he had to formulate all the statements inwords. Thanks to his work, trigonometry became independent of astronomy. Healso compiled tables for the sine function for intervals of one minute with the ra-dius 60,000. He could not free himself from understanding the sine as a half ofthe length of the chord for double angles. Values understood in this way, however,depend on the length of the radius. 6

Voskovec and Werich, in their song from the play T¥ºká barbora (Heavy Bar-bara), were rather uncomplimentary about the Middle Ages. Also in this period,however, science developed, although especially in the narrow circle of educatedmembers of the church. Mathematics usually did not exceed the level set bythe Ancient Greeks, but in contrast to them, people in the Middle Ages valuedmathematics especially for its practical applications and its development was to acertain extent in�uenced by the development of production forces. For more de-tailed treatment, I recommend the readers to consult [Be2], where we can �nd moredetails about some important mathematicians, extracts from their works includingcommentaries, and matters of interest concerning the science and education in theMiddle Ages.

6Unit radius was introduced by Euler only in 1748.


Chapter 4

Mathematics in the 16th and

17th Centuries

Historians have not agreed about when the Middle Ages end: sometimes, it isstated that it ends with the discovery of America, other times, with the inven-tion of printing. Both these events were signi�cant milestones in the history ofmankind, especially the latter. Gutenberg's invention made it possible that ed-ucation gradually ceased to be the privilege of a small number of people. Thediscovery of the new continent then inspired voyages across the ocean and thetrade signi�cantly increased. For travels across the ocean, bigger ships were nec-essary as well as more reliable navigation. The legendary voyage of FerdinandMagellan proved that the Earth is round, Copernicus came with heliocentric the-ory, which was further improved by Kepler and Newton then showed that it hasto be so, from the point of dynamics.

These and other circumstances also served as an impulse for the developmentof mathematics. It now not only could compete with the knowledge of the An-cient nations, but also surpassed it and its development proceeded forward in anunstoppable manner. Complex numbers were discovered, foundations of the in-�nitesimal calculus were established, the discovery of logarithms signi�cantly sim-pli�ed especially calculations in astronomy, probability calculus brought chance tomathematics, algebraic notation signi�cantly simpli�ed the writing of mathemat-ical texts, and we could proceed further in this manner. We will deal in greaterdetails with those areas that can be used in school mathematics.

4.1 Algebraic equations, complex numbers

In one of the �rst printed mathematical books, Summa de Arithmetica, writtenby the Franciscan Luca Paciolli, we may learn, among other things, that solvingof cubic equations is, given the state of mathematical knowledge of that time,approximately as impossible as squaring the circle. Already Ancient Babylonianscould solve some special kinds of cubic equations, partial success was reached by

35

36 CHAPTER 4. MATHEMATICS IN THE 16TH AND 17TH CENTURIES

the Ancient Greeks or the Arabs, but the general formula, or better expressed,and the general procedure was not found. Said with the occultist cook Jurajda,the mathematicians in renaissance Italy were predestined to �nd this solution.

The �rst of them was Scipione del Ferro, who lived in Bologna from 1465to 1526. He found the solution for some kinds of equations,1 however, he didnot publish his solution, only communicated it to a few of his friends. Venetiancomputer Niccolo Fontana (1499?-1557), known under his nickname Tartaglia, inEnglish the Stammerer, allegedly could solve all types of equations, but he did notpublish his results either and only shared them, under the oath of keeping silent,with the Milan physician Geronimo Cardano ((1501-1576). This man, apart frommedicine, devoted his time also to mathematics and contrary to his colleaguesnamed above, he decided not to keep silent about the solution and he publishedhis results and those of his collegues in an outstanding book Ars Magna, in whichhe explained the methods of solving equations of the third and also of the fourthdegree. Contrary to some thinkers north of the Alps, Cardano honestly statedwhich results were his own and which should be attributed to his colleagues. Inthis book, we can thus read that Tartaglia shared the method for �nding rootsof the equations of the third degree, but without a proof, which motivated himto �nd the proof and publish it. This procedure, which seems to us very honest,however, made Tartaglia very angry and a quarrel started between both men, inwhich neither of them stayed away from rude words. In any case, the formulasfor solving equations of third degree are called Cardano formulas in contemporarybooks.

These formulas are hardly used nowadays, we will nevertheless take up somespace to present them here. If the equation x3 + a1x

2 + a2x+ a3 = 0 is given, wecan eliminate the quadratic member through substituting for x = y− a1

3 . We canthus only consider equations in the form x3 + px + q = 0, whose roots are givenby the formula

x =3

√√p3

27+q2

4+q

2−

3

√√p3

27+q2

4− q

2.

When solving the equation x3− 6x+ 4 = 0, we will see that it does not have asolution in real numbers, because the expression under the square root is negative.However, without using any specialised methods, we can con�rm that the number2 solves the equation. Mathematicians solving these equations then could notexplain this paradox and chose to call it casus ireducibilis. Such a case occurswhenever the equation has three di�erent real roots. We can easily con�rm thatthe given equation has also two irrational roots, −1±

√3. Cardano also considered

negative roots, which he called �ctitious. In his work, he also introduced methodsfor solving equations of the fourth degree, which was a discovery of his pupilLodovico Ferrari (1522-1565).

To all probability, casus ireducibilis instigated the origin of complex numbers,because it felt that even a square root of a negative number could have some mean-

1As negative numbers were not fully accepted then, and mathematicians wrote the equationsin such a way that the coe�cients be positive. They did not use contemporary notation either.

4.1. ALGEBRAIC EQUATIONS, COMPLEX NUMBERS 37

ing. Already Cardano suggested this when he published the following problem,which we will state using today's symbolic: If we are to divide 10 into two partswhose product is 30 or 40, it is clear that this case is impossible. Let us, however,proceed in this way: if we divide 10 into halves, a half is �ve; �ve multiplied byitself is 25. If we then subtract from it the required sum, say 40, (-15) remains; ifwe take a square root of this and add it to 5 and subtract it from �ve, we will gettwo numbers which, multiplied by each other, will give 40; these numbers are 5+

√5

and 5 −√

5. It is not clear how to explain the appearance of this problem, andresearchers are inclined to think that Cardano wanted to illustrate by this examplethat even when solving quadratic equations, we might need to calculate the squareroots of negative numbers. If it is so, the thought was revolutionary, because forthe whole three millennia when quadratic equations were solved nobody thoughtthis way. As we can easily see, we could solve the problem as a quadratic equationx(10 − x) = 40. He called square roots of negative numbers quantita sophistica,but he did not get far in determining their properties. Rafael Bombelli (1526-1572) was a di�erent kind of man, because he himself stated eight basic rules forcalculating with a complex unit, or, more accurately, he stated how a unit andan imaginary unit should be multiplied and how to multiply imaginary unit withitself. It is obvious from the formulas he published in his L'algebra parte maggioredell Aritmetica that he already understood complex numbers as we understandthem today. As an example, we can give the equality 3

√52 + 47i = 4 + i.

Complex numbers were relatively slowly implemented in mathematics, as theirinterpretation was not clear. The well-known mathematician and philosopherDescartes was the �rst one to use imaginary as the name of the number, fromwhich later denoting the complex unit as i arose, introduced by Euler. He was theone who started expressing complex numbers in goniometric form and is also theauthor of the well-known identity eiϕ = cosϕ+ i sinϕ. That seems to suggest thathe understood complex numbers as points in the plane, but he did not publish thatexplicitly. The �rst one to understand complex numbers de�nitely as points in theplane was the Norwegian cartographer and geodesist Caspar Wessel (1745�1818),main share on spreading this idea is attributed to Carl Friedrich Gauss (1777-1855) who in his dissertation used complex numbers to prove the fundamentaltheorem of algebra and in this Theoria residuorum biquadraticorum gave geomet-rical interpretation of these numbers in the way we understand them today. It canthus be said that naming the complex plane after this famous scientist is justi�ed.That, however, is not true for the equality (cosα + i sinα)n = cosnα + i sinnα,which is nowadays known as Moivre theorem. The French Hugenot Abraham deMoivre (1667-1754), who spent most of his life in England, did indeed work withcomplex numbers and used this inequality, but he never wrote it in this way. Let usremember also Lazare Carnot (1753-1823), who christened these strange numberswith the nowadays used name complex.

From this review, it is apparent that complex numbers were successfully usedin solving (not only) mathematical problems, although the theory around themhad not been elaborated yet, As an example, we can mention solving linear di�er-ential equations with constant coe�cients, for example the equations describingthe behaviour of a simple electrical circuit with an inductor and a capacitor has


the following form: Ld2Idt2 + 1

C I = 0. Characteristic equation has complex roots,so the solution of this equation is therefore I = C1 cos 1

LC t + C2 sin 1LC t. We can

determine concrete values of these constants, when the initial conditions are given.It is obvious that the dependency of the current on time is given by harmonic func-tion and that harmonic oscillation takes place. This procedure may of course beapplied also to mechanical oscillation. Solving of di�erential equations is usuallynot taught at high schools, but we can also describe what happens in alternatecurrent circuits, like determining the impedance and phase di�erence (phase o�set)between the electrical current and voltage.

4.2 In�nitesimal calculus

This notion stands for two operations, and those are the derivative and the op-eration in some sense inverse of the derivative, the integral. In English, theseoperations are usually covered under the name of calculus. While some parts ofmathematics originate only in the modern times, the integral in particular origi-nated in the Antiquity. The integral namely allows us to determine the area of ageometrical form with curved lines as its boundaries; the problem is thus a ratherpractical one. As we have already said, already the Ancient mathematicians couldcompute the area and volume of some such formations. They did not use theconcept of function, but the basic idea of the integral, namely the division of thesolid into parts whose volume we can compute and then adding these up, can befound already in their work.

Let us have a look at how Pierre Fermat approached the problem of computingthe area of a part of a parabola given by the function y = x2 when x ∈ [0, 1].He took any number q ∈ [0, 1] and constructed the sequence of points 1, q, q2, q3,etc. on the x axis. In these points, he constructed lines perpendicular to the xaxis, intersecting the parabola at points [qi; qi

2]. He then started to circumscribe

rectangles whose one side were two neighbouring points on the x axis and the otherwas the corresponding value on the y axis. When we add up the area of all theserectangles, we obtain

1 · (1− q) + q2(q − q2) + q4(q2 − q3) + . . . =

(1− q) + q3(1− q) + q6(1− q) + . . . =1− q1− q3

=1

1 + q + q2

The closer q comes to number 1, the closer the area of the relevant rectangle isto the area of the parabola and also to 1

3 . From the above-stated procedure, it isclear that Fermat use the approach of limiting sequence, although only intuitively,this notion will still wait to be discovered for another two centuries.

Bonaventura Cavalieri (1598-1647), one of the best disciples of Galileo Galilei,chose a di�erent approach. Cavalieri used the method of cross sections, i.e. cut-ting the plane into segments of in�nitely small width; in the concrete case of theparabola, these parts have the length x2 and in�nitely small width. Cavalierisought for the sum of these squares in the triangle ACE, while he supposed that

4.2. INFINITESIMAL CALCULUS 39

all the transversals with the side AC are exactly those squares. Then∑EA

XY 2 =∑ED

XY 2 +∑DA

XY 2,

where D is the midpoint of line segment AE. Let us choose any point X on theline segment DA. The perpendicular line from point X to the leg AE will intersectthe midpoint transversal in point Z and the hypotenuse in point Y . It the holdsthat∑EA

XY 2 =∑ED

XY 2+∑DA

(XZ+ZY )2 =∑ED

XY 2+∑DA

XZ2+∑DA

ZY 2+2∑DA

XZ.ZY.

The �rst and the third sums are basically the same, because it determines thearea of two equal triangles. In that way, we dealt with uneven lines and we mayconsider the remaining two sums to be normal forms, and thus we will in the endobtain the following formula:∑

EA

XY 2 = 2∑ED

XY 2 +1

4.

Cavalieri stated that the volume of two solids of the same height is the same, iftheir cuts in the corresponding heights are the same. This principle is nowadayscalled Cavalieri principle. According to this principle, we can sum up the areasand we will get the solid, the sum of squares will thus be a pyramid. We canimagine that the individual cuts are in fact in�nitely thin desks. It then holdsthat ∑

EA

XY 2 = 8∑ED

XY 2 = 2 · 1

8

∑EA

XY 2

and after simpli�cation we obtain∑EA

XY 2 =1

3.

Using contemporary terminology, we could say that for determining the in-tegral, Cavalieri did not use the notion of the limit, but of an in�nitely smallquantity. Cavalieri extended his procedure up to 9, and he thus knew the formula∑1

0 xn = 1

n+1 , while this sum is in fact the integral from 0 to 1. Already GalileoGalilei, Cavalieri's teacher, worked with in�nitely small quantities when statinghis law of free fall. He knew that speed is directly proportional to the time, i.e.v = gt. We are asking about the length of the trajectory the falling object withinthe time period t. In a certain point in time t0, the object moves with speed gt0and in an in�nitely small length of time dt, it crosses the distance gt0dt, and in theperiod of time t− t0, the distance g(t− t0)dt. The overall length of the trajectoryis gtdt, and we would reach the same result if the object moved at both pointswith constant speed gt

2 . The length of the trajectory is thus s = 12gt

2. Puritansmight feel that this problem does not belong there, but already Archimedes was


inspired in many of his mathematical discoveries by physic and the same holds inthe 17th century.

Finding the minimum and the maximum of a function was also an importantproblem. In connection with this, let us mention Fermat's work Methodus adDissquirendam et Maximam et Minimam. We will demonstrate Fermat's approachto �nding the tangent to the curve y = f(x) again on the example of the parabolay = x2 in point T = [x0;x20]. The tangent will intersect the x axis in point P andwe need to �nd its x-co-ordinate. If we increase this value by dx, then we willreach point P1 on the tangent and the triangles will be similar. If, in addition, dxis very small, we can substitute the curve by a line segment. From the similarityof triangles it follows that

PQ =dx.x20

(x0 + dx)2 − x20.

Fermat then used the not quite accurate trick of putting the square of dx equalto zero, but considering the �rst power of dx to be non-zero and divided by itboth the numerator and denominator, so the x co-ordinate of the intersection ofthe tangent with the x axis is x0

2 and the slope of the tangent is k = 2x0. If we usecontemporary terminology, we could say that Fermat substituted the increment ofthe function by its di�erential. Although this approach was not entirely correct,it was used by many scientists and led to correct results. Mathematical puritansdid not like this and they protested against this vehemently, but we will describethis later in the sections dealing with crises in mathematics.

Newton allegedly said that he could see further because he was standing onthe shoulders of giants. In the case of in�nitesimal calculus, we can understandthis in such a way that he drew upon the work of all the people who dealt within�nitesimal calculus before him (and we have not mentioned all of them in thisbook). If we nowadays ask who Newton was, then the majority of those whorecognize the name will say that he was an English physicist. This scientist really�nished the edi�ce of classical mechanics and his Principia2 are as important forphysics as Elements are for mathematics. It is less known that this scientist is alsoto be credited with contributions to in�nitesimal calculus and it probably will notsurprise us that physics played a role therein.

Newton used his knowledge of analytic geometry and he imagined a curve ina plane (trajectory) as a set of intersections of lines parallel with the coordinatesystem that move with speeds x and y and he called these speeds �uxions, whilethe coordinates were called �uents. The following must hold:

x =dx

dt, y =

dy

dx.

Their proportion is the derivative of y with respect to x. Newton then formulatedtwo basic problems:1. When we know the dependence of speed on the trajectory of the mass point,

2Philosophiae Naturalis Principia Mathematica or Mathematical foundations of natural phi-losophy, published 1687.

4.2. INFINITESIMAL CALCULUS 41

we determine the dependence of speed on time.2. When we know the dependence of speed on the time, we will determine thedependence of trajectory on the time.

He solved the second problem through seeking a primitive function, i.e. througha procedure inverse to derivation. As Newton then did not yet use the notion ofthe function, we may suppose that Newton only considered continuous functions.When we consider the narrow connections with the motion of an object, we cannotexpect anything else. If a positive function y = f(x) is given on a closed interval〈a; b〉 and if we can �nd such function y = F (x), for which it holds on a certaininterval that F ′(x) = f(x) in the given interval, then the di�erence F (b)−F (a) isequal to the area of a formation whose boundaries are the x axis, the lines x = aand x = b and by the graph of the function y = f(x). In other words, we de�nethe so-called Newton integral:

(N)

∫ b

a

f(x)dx = F (b)− F (a).

Gottfried Wilhelm Leibniz (1646-1716) chose a di�erent approach. A youngman educated in the humanities, he came to Paris in 1672 as a diplomat in theservice of the archbishop in Mainz.3 As a diplomat, he certainly met the Sun KingLouis XIV, but it was his meeting with Huygens. It was under the in�uence ofthe Dutch scholar that he converted to mathematics and among other things hemade himself familiar with Pascal's idea of so called characteristic triangle and hedeveloped this thought, used in a special situation, into a coherent theory.

Let the function y = f(x) be given. Let us construct the tangent to thisfunction through point A and let us construct the perpendicular line to this tangentin point A. The perpendicular line, the x axis, and the parallel with the y axisgoing through pointA determine the characteristic triangleAPR, where | AP |= y,| PR |= m. This triangle is similar to the triangle ABC, where dx =| AB |,dy =| BC | are in�nitely small. From the similarity of these triangles, we candeduce the equality mdx = ydy. We can proceed in this manner in any point andadd the results. Leibniz, however, tried to compare the curves y = x3 and y = x2,and he obtained the result

∫x2dx = x3

3 . He found the integrals of other curves inthe same way.

Both Newton and Leibniz worked independently of each other and their resultsare equivalent in some ways and di�erent in others. They both discovered theconnection between the derivative and the integral and they derived basic formulas,and although they drew upon the work of their predecessors, they enriched theirwork signi�cantly. If we used some kind of a metaphor, we could say that they�nished the basic edi�ce of the in�nitesimal calculus. Newton, however, solvedmainly concrete problems and he was often inspired by physics. Leibniz, on thecontrary, worked in a more mathematical manner, was looking for general methodsand algorithms and he tried to unify the approach to various problems. Newton'ssymbolism was rather awkward and impractical. Using a dot above a letter beforeTeX was invented was rather courageous. On the other hand, Leibniz devoted

3The archbishop in Mainz was also one of the prince-electors (Kurfürst) of the Roman king.


lots of attention to symbolism he used and we use his symbolism until today. Theone thing that was common to both gentlemen was their loose treatment of thein�nitely small quantities. We will soon see what came out of that.

Both men also entered a quarrel over the priority of their invention. As we havealready stated, they did not invent the theory, but their contribution was central.Newton started to work on these problems earlier then Leibniz, but he publishedhis results with a signi�cant delay. In this sense, Leibniz was a scientist of the 21stcentury already and he was getting points by publishing his results early. In thesecond half of 17th century, in addition, there were no scienti�c journals and topublish the results meant to publish an independent book and their spread beyondthe boundaries of the country of origin was an exception. There were no scienti�cconferences, and the exchange of �ndings between scientists was through letters,if they corresponded at all. Therefore we can conclude this discussion by sayingthat they both arrived to their results independently of each other.

4.3 Probability theory

The date of birth of this discipline could be determined to be 1654 m when fre-quent correspondence on hazard began between Fermat and B. Pascal. Just likePilatus became part of a credo, a knight Antoine Gombaud de Méré became partof mathematics. This French nobleman liked hazardous games and although wecannot deny that he did have some mathematical knowledge, he could not grapplewith some problems, and therefore he asked Pascal for help and Pascal consultedhis solutions with Fermat in a letter exchange. Their primary interest was espe-cially the division of a bet among two equal players, who could not �nish theirgame for some reason. Let us suppose that it is as if in the play o� part of thetournament, the rules would require four victories (it is not quite clear from thesources what kind of game it was), and when the intermediate result was 2:1 infavour of player A, they had to prematurely end this series. The honour of theplayers required that the bet be divided and it was also clear that player A shouldget more, but they could not agree on the ratio.

Fermat realized that the whole series must end after four games. If thereare four more games, then there are 16 di�erent sets of results of those games.(Variations of the 4th order with two elements: V2(4) = 24 = 16.) It is thereforeenough to write down all the possibilities of the continuation of the series anddetermine how many times player 1 would win the bet and the problem is solved.If we do this, we �nd out that in 11 cases, player A would win and in the remaining5 cases, player B would win. It is therefore necessary to divide the bet in the ration11:5 in favour of player A. Fermat correctly realized that it is necessary to takeinto account all the possibilities, including those that would in reality not happen,because the series would have been won before. There is a certain similarity withpenalties in football matches. The rules prescribe �fteen scores or each side, andthus we have to take into account all the possibilities 45. A minor remark toend this section: as late as a hundred years later, d'Alembert thought that whentossing two coins, there are only three results possible (heads-heads, tails-tails,

4.3. PROBABILITY THEORY 43

and heads-tails). It should not have been di�cult for him to take two coins andillustrate the results.

Pascal developed similar ideas, and he expressed in his letters that he wascontent with the fact that the truth was the same in Toulouse and in Paris. Pascal,however, developed his thoughts even further and he reached a general solutionwhich can be formulated in the following way: Let m victories are lacking forplayer A to win and player B is lacking n victories to win. Then the number ofseries that need to be played is at most m + n − 1. Player A will win the bet ifhis rival wins at most n − 1 games. Player B has a chance, if player A only winsm− 1 times. It is therefore necessary to divide the bet in the following ratio:

n−1∑i=0

(m+ n− 1

i

):

m−1∑j=0

(m+ n− 1

j

).

It is a pity that this problem does not appear much in textbooks, while theauthor used it with a lot of success when he was teaching at high school as amotivation problem when introducing theory of probability. More details, and notonly about this problem, may be found in [Ma2].

In those times, correspondence belonged to common methods of scienti�c work,so the problems attacked by Fermat and Pascal were communicated also to otherscientists, for example to Christian Huygens. 4 This excellent persona of worldscience contributed also to probability theory by publishing his work De ratiociniisin ludo aleae. This work is not very extensive, but is well worth attention. Itcontains 14 themes that are called Propositio. Let us state the �rst three of them:

P1: If I expect sum a or sum b, which I can both obtain with the same e�ort,then the value of my expectation is a+b

2 .P2: If I expect sum a, sum b or sum c, which I can all obtain with the same

e�ort, then the value of my expectation is a+b+c3 .

P3: . If the number of cases in which I expect sum a is p and the numberof cases in which I expect sum b is q, and if I assume that all cases can happenequally easily, then the value of my expectation is pa+qb

p+q .In these three de�nitions, expected value is de�ned for the �rst time, although

Huygens does not state this explicitly. As the book deals with games, he usesthe term expected winning. Similarly, he does not use the term probability andinstead says that all the results can be obtained equally easily.

Propositiones 4-9 solve the problem of dividing a bet. Huygens starts withsimple examples and proceeds to the more complex ones and he also adds players.We will only state the full version of Propositio 9, as in that one, Huygens statesthe general method for solving the problem, by which he reached a level higherthan Fermat or Pascal who only solved concrete problems.

P9. In order to calculate the share of each player in a game with an arbitrarynumber of players, from whom some lack more and other lack less victories, wemust �nd out what belongs to the player whose share should be determined when hehimself or somebody else wins the next game. By adding the parts obtained in this

4Christian Huygens (1629-1695), Dutch scientist who lived in Paris for a part of his life. Hediscovered, among other things, the rings of Saturn and formulated wave theory of light.


way and dividing it by the number of players, we obtain the share of that particularplayer.

A table presenting the solution of the problem for 17 situations that can happenin a game of three players is attached to this Propositio. The remaining Propositiaconcern the game of dice. At the end of the book, we �nd �ve unsolved tasks(Problemata). We refer those who are interested in more detail to book [Ma2].

When dealing with mathematics of 17th and 18th centuries, we can be surethat we will encounter the name Bernoulli. Probability is no exception. We needto mention especially Jacob I (1654-1705), the author of Ars Conjectandi. Thisbook remained incomplete, it was published only in 1713 by Jacob's son NicolausI. In the �rst part of the book, we will �nd the full text of Huygens's De raciociniisin ludo aleae including detailed commentary. Second part is basically a course incombinatorics and the third part draws on to the second, as combinatory problemswith the theme of games. In the last, unfortunately un�nished, part, we can also�nd the �rst formulation and proof of the law of large numbers, which is probablyJacob Bernoulli's greatest contribution to probability theory.

High school students who have probability and statistics included in their schoolcurriculum will probably meet the name of Bernoulli in connection with binomialdistribution. Let us perform n independent attempts, while the probability thatthe phenomenon A occurs in one of the attempts is p and it is the same for allthe attempts. Then the probability that the phenomenon A occurs in this seriesexactly k times is given by the formula

pk(A) =

(n

k

)pk(1− p)n−k.

This arrangement is sometimes called Bernoulli's scheme or Bernoulli sequence ofindependent attempts and it provides a wide opportunity to present problems inprobability. In any case, the following problem should be presented to the students:

The test consist of 20 questions, and for each questions, four possible answersare given, of which only one is correct. A student did not prepare for the test and�lls out the answers at random. What is the probability that he will pass the test,if he needs to answer 15 questions correctly for that?

The probability that a high school student will meet the name of Bernoulliin connection with another result in probability theory is not very high, whichis de�nitely a shame. Jacob's nephew Daniel published the following problemincluding its solution and as it happened in St Petersburg, it is known as theSt Petersburg paradox.5 The author of this book never forgot to mention it tostudents, it was very well received, but none of his students ever solved it. What,then, is St Petersburg paradox?

Two players, of whom one is a banker,6 play the following game: the bankertosses a coin and if it turns out to be heads, the game ends and the banker will payhis rival 2 crowns. In the opposite case, the game continues with another toss and

5This problem was only published in the 18th century, but as we will not be mentioningprobability theory in the following text again, we decided to present it here.

6Here, we do not have in mind the owner of a �nancial establishment, but the person acceptingbets and giving away the winnings.

4.4. RUDOLPHINE PRAGUE 45

it the result is heads, the game ends, but now the banker has to pay 4 crowns. Thegame goes on until the banker does not succeed in tossing a heads. If this happensin the n-th attempt, the banker will pay his rival 2n crowns. The question is, howmany crowns the player must put into the game so that the game is just, i.e. sothat the victory or loss of one of the players was only a matter of chance.

As has already been said, when the author presented the task to his students,none of them solved it, although the solution is not di�cult and the necessaryformula had been taught before. The expected value of the winning is namely

E =

i=n∑i=1

piEi =1

2· 2 +

1

4· 4 +

1

8· 8 + · · ·

In order for the game to be just, the player would have to start by paying in�nitelymany crowns. This was a great surprise for the students, especially when they triedthis game before the calculation. Calling it a paradox is thus fully justi�ed.

If the students have already been taught about geometric series, the teachercan return to this problem again. The paradox consists in the fact that the bankerhas to pay any amount of money the player wins. Geometric sequence is tricky inthis sense, as one Indian maharaja who could not pay the reward he promised tothe inventor of chess could con�rm, as well as those who have trusted a fraud inpyramid scheme type of game. We therefore adapt the game slightly by determin-ing that the highest winning to be paid out is 64 crowns. The expected value ofthe winning will thus be

E =1

2· 2 +

1

4· 4 +

1

8· 8 +

1

16· 16 +

1

32· 32 + 64

(1

64+

1

128+

1

256+ · · ·

),

i.e. roughly 5.03 crowns. Even in this case, however, we need to take into accountall the possibilities, even if the game would in reality end with the �fth attempt.

Z uvedeného textu by se zdálo, ºe i tak serózní v¥da jako je matematika nebylaimunní v·£i gamblerství a bezuzdn¥ mu propadla. Jenºe tomu tak není. Teoriepravd¥podobnosti na²la záhy své místo v pojistné matematice, nebo´ na zák-lad¥ statistických údaj· (úmrtnostních tabulek), lze stanovit pravd¥podobnost,ºe se daná osoba doºije v¥ku v. Spojení teorie pravd¥podobnosti a statistikyje p°irozené, na pravd¥podobnosti je zaloºena i moderní fyzika, p°esn¥ji °e£enokvantová mechanika. Záv¥rem povídání o pravd¥podobnosti p°ineseme n¥kolik za-jímavostí z na²ich luh· a háj·. Jedním z prvních na£ich matematik·, kte°í se zabý-vali touto disciplínou, byl Václav �imerka7, který se snaºil pomocí pravd¥podob-nosti °e²it n¥které �lozo�cké otázky. [Sm2].

4.4 Rudolphine Prague

Also this chapter will be closed with a few remarks about mathematics in thelands of the Bohemian crown. This time, this section will not be short and,

7Václav �imerka (1819�1887), kn¥z, u£itel a matematik.


in particular, it will not concern only matters of regional interest. In 1576, theemperor Maxmilian II died and the heir to the throne was his oldest son, Rudolph,second of that name among Habsburg kings. Bohemian noblemen then performeda daring escapade and succeeded in convincing the young emperor to move toPrague. Prague agglomeration thus became the centre of the central-Europeanunion of states and its fame again reached the stars again after more than twocenturies. The emperor had somewhat weird habits, but arts and sciences werehis passion and Prague became a favourite destination of scientists as well as"scientists". We will not pay attention to the latter ones, although, according tothe well-known movie directed by Fri£, they indeed invented some good thingslike glue, �oor polish, and plum brandy, for example. We will rather note the realscientists, both local and from abroad, who were attracted by Rudolph's generosity,although there often was a signi�cant di�erence between the promised and receivedsalary.

In the �rst place, we need to mention the astronomer, botanist, and personalphysician of the emperor, Hagecius (also known as Tadeá² Hájek of Hájek or Thad-daeus Hagecius ab Hayek, 1525-1600). It was Hagecius who invited the Danishastronomer Tycho Brahe(1546-1601) to Prague. Tycho Brahe received the castlein Benátky nad Jizerou, where he transferred his astronomical observatory. Brahewas an excellent observer and he constructed and improved many astronomicalinstruments, but he did not ask the emperor for lenses, since the telescope wasonly invented several years after his death. When Brahe died, he left lots of ma-terial about the movement of celestial bodies that he did not manage to process.This task was taken up by the court mathematician of Rudolph, Johnanes Kepler(1571-1630). Kepler focused in particular on the movement of Mars and the morehe looked into the data, the more he came to realize that it will be necessary toleave the thousand years old idea of circle as the most perfect of all curves andthat therefore all celestial bodies must move along circular trajectories. Thanks tothis revolutionary thought, it was possible to put the heliocentric theory of Coper-nicus in accord with the movement of the planets in reality. Ptolemy's geocentricsystem, although it was very complex, with all the epicycles that had to be addedall the time, was namely giving better computational results. Kepler's Astrono-mia nova was very important for further developments in astronomy, as his lawsdescribed the movement of big celestial bodies. Kepler succeeded in describing themovement of the objects in the sky, but he could not explain why it is so. Thatwas only achieved by Newton, who discovered the general law of gravitation.

Let us shortly remind ourselves of his laws. The �rst one says that the planetsmove on elliptical trajectories in the centre of which is the Sun. The second lawsays that a line joining a planet and the Sun sweeps out equal areas during equalintervals of time (areal velocity is the same). Thanks to this law, the summeron the northern hemisphere is by several days longer than the summer on thesouthern hemisphere. When there is summer in our country, the Sun is in aphelion(farthest from the Sun) and it therefore moves more slowly. The Russians senttheir telecommunications satellites on trajectories with big eccentricity and thesethen stayed for a long time above their territory. Kepler later added also anotherlaw, saying that the square of the orbital period of a planet is proportional to the

4.4. RUDOLPHINE PRAGUE 47

cube of the semi-major axis of its orbit. To end this passage on Kepler, we needto correct the information given in the famous �lm directed by Fri£. It was notTycho Brahe, but Kepler who was the professor who asked the emperor for a lensand his request was ful�lled. Kepler already used a telescope and, contrary toGalilei, his eyepiece had a lens.

Kepler also belonged to the pioneers in using algorithms. His tables had beenassembled by his assistant and a clockmaker from Switzerland, Joost B¶rgi (1552�1632)


Chapter 5

From 18th century till now

The history of humankind in this period is full of revolutionary events and lifechanged signi�cantly in it. Just as technology was developing fast, mathematicsalso had to be developing fast. We will, paradoxically, shorten our account ofthis period, because the results are glamorous, but they exceed the framework ofmathematics taught at elementary schools and high schools. Those interested inthe development of mathematics in this period are thus advised to look into othersources.

5.1 The enlightened century

This section is named after the movement that characterizes this century. However,as we are writing about the queen of the sciences, the title Euler's century wouldbe more appropriate, after the mathematician Leonhard Euler (1707-1783). Ashis name was mentioned in other parts of this book, we will not deal with hispersonality in detail. Bernoulli family, in which the mathematical genes wereinherited so consequently that some of the family members have to be referred toby numbers next to their names, as with kings, also has Swiss roots. BrothersJohann (1667-1748) and Jacob (1654-1705) devoted their attention to analysis,in which they achieved substantial results, which are nowadays part of the mainbody of knowledge called mathematical analysis. The contribution of this familyto probability theory has already been mentioned. Bernoulli equality, known fromhydrodynamics, which is taught in physics, is the discovery of Johann's son Daniel(1700-1784).

In France, which is not far from Switzerland, many excellent mathematicianswere active in this period. Jean Le Rond d'Alembert (1717-1783), from 1754 per-manent secretary of the French Academy of the Sciences, was also a leading �gureamong the encyclopaedists, especially for mathematical part of it. Together withDaniel Bernoulli, he is the founder of the theory of di�erential equations. He isalso one of successful physicists (hydrodynamics, aerodynamics, three body prob-lem). His mechanical principle, reducing dynamics to solving static problems, iswell known. He introduced the notion of the limit. He also devoted some atten-

49

50 CHAPTER 5. FROM 18TH CENTURY TILL NOW

tion to probability theory, but was not very successful. We know his "paradox",in which he mistakenly determined the probabilities for tossing two coins. He saidthat the probability that the heads falls at least once is 2

3 , because he consideredthe possibilities HH, HT, TT instead of the correct HH, HT, TH, TT. The correctresult therefore is 3

4 .Joseph Louis Lagrange (1736-1813) was born in Turin and was of French-

Italian origin. He worked in Turin, Berlin, and Paris. To begin with, he devotedhis time to the calculus of variations and he discovered many original results inthis �eld. Apart from that, he also ordered and compiled historical material,which was typical for him also in other �elds of his interest. He then appliedhis discoveries in mechanics, for example, he provided the �rst particular solutionof the three body problem. He summed up his results in his book Mécaniqueanalytique, in which he united the di�erent principles of statics and dynamics. Inthe in�nitesimal calculus, he introduced an algebraic approach when starting fromthe development of functions in Taylor series. Let us mention, as a curiosity, thatthe symbols for derivation that we use nowadays are the work of this particularman. During French revolution, he was the president of the measures and weightscommittee.

Pierre Simon Laplace (1749-1827) published two big pieces of work: Théorieanalytique des probabilités and Mécanique céleste. In the second book he summa-rized the contemporary knowledge of mechanics. We have already spoken aboutNapoleon's relation to mathematics, so we cannot be surprised that he was alsointerested in this big volume and he wondered why it did not mention God any-where. Laplace answered: "Sir, I did not need this hypothesis. " We however, haveto remind the reader that Laplace had good relations also with the Bourbons: hisshort episode as a minister in the times of Napoleon's rule was forgiven, and inaddition, he did not prove himself to be a good minister. The worse politician,the better scientist he was and his work inspired many other scienti�c colleagues.Thanks to him, the name of Bayes was not forgotten, and thus we today use theformula named after this English priest for calculating probability a posteriori.

In the Czech lands, mathematics was mainly devoted to applications. A typicalexample of this is the work Solid beginning of the art of mathematics by WenceslasJoseph Weselý, a land miller and geometer, in which we can �nd examples ofpractical surveying including trigonometry. Other books were devoted to practicalor merchant mathematics, calculating the interests, and other problems. Only inthe second half of the 18th century did Czech science and thus also mathematicsstart to make up for the delay. Joseph Stepling (1716-1778) published his bookExercitationes geometrico-analyticae in 1751, which is devoted to the integration ofcertain analytic functions and which raised some interest and was soon publishedtwo more times. In 1765, he then compiled a work on di�erential calculus, in whichhe also tried to extend Euler's thoughts. He is also to be credited with building theobservatory in Klementinum in Prague, which can boast with the longest sequenceof meteorological observations in Europe, having started them in 1775. In 1780,Newton's Principia was published, to which Jan Tesánek (1728-1788) wrote anintroduction. In this work, he tried to make the notion of limit more exact andthus to link this theory to d'Alemberts. He was also one of the �rst number

5.2. THE CENTURY OF STEAM 51

theorists in the Czech lands and he tried to solved the so-called Pellé's equationdu2 + 1 = v2. Franti²ek Josef Gerstner (1756-1832) is probably the best knownname from this period: he designed the known railway with carriages drawn byhorses from Budweis to Linz. That proves that his domain was rather applicationof mathematics in practice. Around 1770, the Learned Society was founded, whichlater developed into Royal Bohemian society of arts and sciences.

5.2 The century of steam

This is the traditional name for the 19th century, and it probably should signifythat there was a signi�cant development in the industry and in agriculture, andthat the symbol and motor of these changes was the steam engine. Science, in-cluding mathematics, also had to comply with the new situation. The age of thetitans who mastered various and often very di�erent branches of science slowlyends and the time of specialization comes, also in mathematics. The practice ofmathematics moves to schools, and mathematicians, apart from doing research,devote more and more of their time to teaching mathematics. This century alsobrought new branches of mathematics.

One of the last scientists of the old age was Carl Friedrich Johann Gauss(1777�1855).

Geometry witnessed a major development. Gaspar Monge (1746-1818) taughtat the military academy in Méziers on the construction of fortresses and thanks tothis, he began to develop descriptive geometry as a special �eld. When studyingspace curves and surfaces, he started using in�nitesimal calculus, by which he es-tablished another �eld - projective geometry. His thoughts were further developedby Victor Poncelet (1788-1867) in his Traité des propriétes projectives des �gures,in which all the basic notions of this �eld are included.

The birth of new �elds, however, was nothing compared to what Euclid causedand what surfaced exactly in this century. Euclid formulated his postulates ingeometry in his �rst book and while the �rst four are formulated in a very simpleform, the fourth one is rather clumsy and talkative; you may judge it for yourselves:

1. I can draw a straight line from any point to any other point.

2. And [I can] prolong the bounded straight line without a break.

3. And [I can] draw a circle with any centre and with any radius.

4. And all right angles are mutually equivalent

5. And should a line fall on two other lines in such a way that the inner anglesalong one side added together are smaller than two right angles, then if weprolong such lines unboundedly, they will meet on the side where the anglesare smaller than two right angles.

This disparity and di�erence in formulation was noticed already by Ancientscientists and thus they had the idea of proving this postulate, whether it is not a

52 CHAPTER 5. FROM 18TH CENTURY TILL NOW

mathematical theorem.1 Ptolemy, Násiruddín Túsí, Lambert , and Legendre. The�rst of them lived in the Antiquity, the second in the Middle Ages, the last two inthe 18th century, and thus we can see that the problem troubled mathematicianspractically continually for over two millennia. Neither those whom we named, norothers who tried to tackle the problem, however, found the proof. Not even Gausswas successful, but he came up with the idea that this postulate is independent ofthe others and that therefore it is possible to leave it out or to substitute it with adi�erent one. However, this thought felt too daring to him and Gauss kept it forhimself, but luckily, he was not the only one who developed thoughts in this wayand these other two gentlemen had the courage to publish their idea. The �rst onewas the Hungarian János Bolyai (1802-1860) and the second the Russian NikolajIvanovich Lobachevsky (1793-1856).

1Later, it was found out that the fourth postulate could also be proved, but this fact did notcause any hassle.

Chapter 6

Three crises in the history of

mathematics

6.1 The �rst crisis

We have already suggested some causes for this crisis in the section on Pythagore-ans, but the discovery of irrational numbers was not its cause. We can mentionZeno of Elea (490-430 BC) and his famous aporias. Originally, there were around40 of them, but only 9 were preserved until today and it is questionable whetherthey are original or whether they have been distorted during the copying of thetexts. The most famous aporia is the one about Achilles and the tortoise, whichwe do not even have to repeat here. Let us mention also the one about the �yingarrow, the one that denies the existence of movement. If, Zeno says, we look at a�ying arrow, we only see it because it at that very point stays in the same place.Motion therefore consists of a number of motionless moments, which is not possi-ble. In the aporia on dichotomy, he then states that we cannot move from pointA to point B, because we �rst have to travel half of this distance, and then halfof this half, etc.

Even a small boy, however, can surpass a tortoise, let alone Achilles, and thearchery masters like William Tell, Robin Hood, or the Native American Indiansproves that the arrow moves. In the same vein, everybody can travel from one pointto another, although sometimes we might prefer this not to be possible. Thesefacts, however, could not confuse Zeno and his argument ran roughly as follows:Yes, it is true that we can see movement, but if we want to understand what theeyes are seeing, we have to approach movement in the way I had suggested. AsI have shown, we cannot think of movement, because movement is only properfor the changing world of the senses, but alien to the real being. In his aporias,Zeno showed the opposite of the continuous and discrete movement, the movementand its re�ection in our minds, or, in other words, the opposition between thecognition through senses and through reasoning. After all, we meet this paradox inmathematics quite often, with non-Euclidean geometry being a beautiful example.

53

54 CHAPTER 6. THREE CRISES IN THE HISTORY OF MATHEMATICS

The Greeks have dealt with the problem of irrational numbers by geometrizingmathematics. We can imagine the square root of two as the diagonal in a squarewith the side equal to one. We can see it clearly in books 7�9 of Elements, whereEuclid understands numbers to be line segments.

6.2 The second crisis

The origins of the second crisis also lie in Ancient Greece, but it only fully grew inthe 18th century. If we simplify it, we could say that this crisis was caused by thein�nitesimal calculus, or, more accurately, the way how this discipline establisheditself. Let us remind ourselves about the way how Archimedes performed squaringof the parabola. In plain language, he �lled the shape with triangles until heconsidered that calculated value accurate enough. Archimedes, however, was arather solitary scientist, but in 17th century, more and more scientists dealt withproblems of this kind. From the examples given it is obvious that they approachedthe problems rather loosely and they did not pay much attention to the details.If we, for example, look at Newton's way of deriving the formula (xn) = nxn−1, itis hardly understandable from today's point of view. Why not, if it is convenient,could we not put an in�nitely small quantity o equal to zero and then neglect it?On the other hand, at other times, we need to divide by this small quantity andin that case, it is inconvenient that it is equal to zero, so we say it is not equal tozero and divide by it as we please.

We may boldly liken the establishment of in�nitesimal calculus to the timeswhen people began to settle in the Wild West. A discovery after discovery wasbeing made, without the mathematicians re�ecting much on the way how theyarrived to that result, what counted was that it gave results that correspondedwell with the reality and helped to grasp it, especially in physics. The procedurefor deriving more and more discoveries was rather formal, and the scientists of thattime had imagination and especially a good nose for reaching good results. Theauthor apologies for this familiar expression, but he has not found a better �ttingexpression in standard language. Euler was a real master in this sense, and we willdemonstrate his skill on two examples - in the �rst case, he arrived to the correctresult, but in the second one, he was wrong, but that was a very rare exceptionin his case, and it only con�rmed the rule that Euler's work is part of the pillars(not only) of analysis.

When teaching complex numbers in high schools, we calculate the powers of acomplex number with the aid of Moivre theorem and the binomial theorem. Eulerproceeded in the same way when he considered in�nitely large n, so that ε = x

nwas in�nitely small. It holds that

(cos ε+ i sin ε)n = cosnε+ i sinnε =

n∑k=0

(i)k(n

k

)sink ε cosn−k ε.

If we for example compare imaginary parts, we obtain

sinx =

(n

1

)sin ε cosn−1 ε−

(n

3

)sin3 ε cosn−3 ε+ . . . .

6.2. THE SECOND CRISIS 55

As ε is in�nitely small, then cos ε = 1 and sin ε = ε. As n is an in�nitely smallnumber, n = n − 1 = n − 2 = . . .. The equality can thus be rewritten in thefollowing form:

sinx =nε

1− n3ε3

!+ . . . =

x

1− x3

3!+ . . .

which is the correct expansion of the sine function into power series. We willobtain the expansion of the cosine function in the same way.

Euler thought that each in�nite sum will be a result of a "correct function".Therefore he thought that

1− 1 + 1− 1 + . . . =1

2,

since it su�ces to substitute in the expansion

1

1− x= 1 + x+ x2 + x3 + . . .

number 1 for x. Guido Grandi, a professor from Pisa, arrived at the same con-clusion, namely by paring conveniently (1 − 1) + (1 − 1) + . . . = 0 and 1 − (1 −1) + (1− 1) + . . . = 1 and using the statistical consideration known from jokes andcomedy scenes.1

It is thus not surprising that some mathematicians did not like the methodsdescribed above and that they started criticizing them. Thus Bernard Nieuwentijdt(1654-1718) criticized Leibniz's approach and he in particular said that higherdegree derivatives do not have any sense. The bishop George Berkeley (1685-1753)then attacked Newton even more directly. In 1734, he published the book TheAnalyst, in which he uses similar arguments as those that we have given earlier.It was not really possible to deny the correctness of the results, but Berkeleythought that the correct results were obtained only because two mistakes basicallycompensated each other. He called in�nitesimal quantities ghosts of deceasedquantities and he said that who will believe in second or third �uxion, or the secondor third derivative, cannot deny any feature of a deity. Berkeley's tractate namelyhas the subtitle. A Discourse addressed to an In�del Mathematician, who wasnobody else but Newton's pupil Edmund Halley (1656-1742). He is the same manwho used Kepler's and Newton's laws to calculate the trajectory of one signi�cantcomet, which he thus ranked among the satellites of the Sun. The comet, bearinghis name, keeps to the timetable he had prescribed for her, unlike trains and otherpublic transport, and every 78 years, it shines in the sky. The story has it thatthe in�del Halley lured away one soul from the bishop, which made the religiousman angry and provoked him to write the book we have just mentioned.

This story may only be well thought-out fable, but it captures the core ofthe problem. Although mathematical tools were created in a strange way, it wasnot possible to deny that they helped in an excellent way the physicists and theastronomers solve problems posed before them Things have reached such a state

1When I eat one chicken and my friend eats nothing, we have both eaten half of a chicken.Grandi was not that down-to-earth and gave an example of two brothers who take turns andhold on to the diamond inherited from their father alternately for a year at a time.


that when Laplace presented his Mécanique céleste to Napoleon, he could answerhis question as to where God is, "Sir, I did not need that hypothesis". It wastherefore not by chance that the opponents of in�nitesimal calculus were recruitedespecially among idealists. Berkeley namely was not only a parish priest, but alsoa well-known idealist philosopher. It was therefore obvious that something needsto be done in this matter.

The �rst problem was the function itself. Albeit founding fathers worked withfunctions, a function to their understanding was some dependence expressed by aformula and functions were of course continuous. The �rst de�nition of a functionin the contemporary sense comes from Johann Bernoulli from 1718: A quantitycomposed in an arbitrary way from a variable an constants is called a functionof the variable. Euler proceeded in the same way, although more accurately:Analytical expression composed in an arbitrary way from variable and constantquantities is the function of the variable. Seven years later, in 1755, he gave thefollowing de�nition: When some quantities depend on others in such a way thatwhen we change those latter ones, the former ones change as well, we say that theformer are a function of the latter. In 1822, Fourier wrote: Generally, functionf(x) is a sequence of values of which each is arbitrary; we do not assume thatthese values are subject to some law, they follow one another in an arbitrary way.Fourier also left the idea that a function must be continuous, although he onlyconsidered a �nite number of points of discontinuity. In 1829, Dirichlet states thefollowing: y is a function of x, if only one single value of y from the interval givenexists for each value of x. He goes on to add that it is not of importance whethersome formula for this exists or not. His de�nition is used until today. He givesan example of a function where y = 1 when x is rational and y = 0 when x isirrational.

The real number itself was another problem. On the other hand, the sets ofnatural, integer, and rational numbers can be de�ned rather easily, be it intuitivelyor with the help of axioms, as Giuseppe Peano (1858-1932) did. He de�ned thenatural number through the following �ve axioms: 1. 1 ∈ N2. Number 1 is not a successor a+ of any number a ∈ N .3. If a ∈ N , then also a+ ∈ N .4. If a+ = b+, then a = b.5. Let S ⊂ N , 1 ∈ S and let for each a ∈ S it holds that a+ ∈ S. Then S = N .(This is the axiom of mathematical induction.)

With these axioms, we can then de�ne the sum and product of natural numbers,integers as a − b, and rational numbers as a

b , i.e. as ordered pairs of naturalnumbers. It holds that the ordered pairs (a, b) and (c, d) determine the same integerif and only if a+d = b+c and these two ordered pairs determine the same rationalnumbers if and only if ad = bc. This approach might look rather complicated to thereader, but the simple de�nition uttered by the German mathematician LeopoldKronecker (1823-1891) was not accepted by the mathematical community at all.2

The construction of real numbers from rational numbers is not easy, but math-ematicians nevertheless succeeded in it around 1870. In this connection, we in-

2This man, who also had a strong in�uence also M. Lerch, claimed that natural numbers arefrom God and everything else is human invention.

6.3. THE THIRD CRISIS 57

troduce the names of Charles Méray (1835�1911), Georg Cantor (1845�1918) andRichard Dedekind (1831�1916).

�tefan Znám made a jesting comparison in his book [Zn] by comparing thechateau of in�nitesimal calculus, constructed over two centuries, to the house ofBaba Yaga. He did not mean its size or its grandeur, but the fact that this chateau,like Baba Yaga's house, stood on a chicken leg. It was therefore necessary toconstruct �rm fundaments for this building. At this point, we will introduce thenames of Bernard Bolzano (1781-1848), Augustin Louis Cauchy (1789-1857), andKarl Weierstrass (1815-1897). The �rst two names are connected with Prague.Bolzano lived in Prague and worked there as a professor. Cauchy then several timeslived in this town with the deposed king Charles X, because he was a teacherof his children. Bolzano's works were, however, not known to the public (if hepublished his thoughts at all). Cauchy's works, on the contrary, were widelyknown, especially his masterpiece Course d'Analyse published in 1821. Cauchycorrectly de�ned elementary notions as continuity (Bolzano could also do that) andlimit, he also uses the notion of in�nitely small quantity, which he further speci�eslater. Cauchy (mistakenly) thought that each continuous function has a derivativeexcept for a �nite number of points. Bolzano was the �rst one to construct acontinuous function which did not have a derivative in any point, but he neverpublished this discovery. The �rm foundations for mathematical analysis werecompleted by the last of the three, who introduced the ε-δ symbolism used untiltoday. His de�nition of the limit created di�culties for generations of students, butit had bene�cial in�uence on the in�nitesimal calculus. Thanks to this approach, itwas possible to remove intuition from analysis and start with proving the theoremscorrectly. This process is sometime called the arithmetization of analysis, whenthe previous dynamical approach was substituted for by static approach. From thepoint of view of mathematics, this approach is faultless, but for teaching at highschools, it is advisable and follows from the author's experience to start explainingthe derivative as Newton did. Dynamic approach is in compliance with practice intechnology, when planning a railway viaduct, only the weight of the passing trainsis taken into account. The fact that a hamster can also appear on the bridge atthe same time with the train is overlooked, although this cute animal also loadsthe bridge.

6.3 The third crisis

This crisis also has its origin in Ancient Greece when Epamenides, the citizenof Crete, disgusted by the dishonest behaviour of his fellow citizens, uttered thefamous sentence that all Cretans are liars. This has really happened, and wasnoted by Eubolides of Miletus, and we will return to this later. This statementseems not to have anything in common with mathematics, since it concerns logic.

The third crisis is due to set theory and the notion of in�nity. Even AncientGreek mathematicians had to grapple with this notion, but they always under-stood it in the sense of potential in�nity, i.e. in the sense that some set does nothave limitation. The second axiom of Euclid states that a line segment (a line)


can be arbitrarily prolonged beyond its endpoints, a statement in Book X thenstates that there are more primes than any given number. In fact, until the 19thcentury, nobody came up with the idea of thinking about in�nity in a di�erentway. The �rst one to think of sets di�erently was the already mentioned Bolzanoin his posthumously (1851) published book Paradoxien des Unendlichen. Bolzanonamely thought of trying to watch the situation from above. In the work justmentioned, he a.o. says: In order to think of a whole (an in�nite whole) composedof objects a, b, c, etc., we need not have an image of each of them in our minds.We can think about the set of all the inhabitants of Prague as a whole without hav-ing an image of each of the inhabitants of the town. Cantor was another one wholooked at sets as a whole, but before we state what he found out, let us present ashort diversion.

Let there be two �nite sets and we should �nd out which of them has moreelements. One of the possibilities is that we count the number of the elements ofthe �rst and of the second set and compare the results. Can also a person whocannot count solve this task? The answer is yes, it the person thinks of pairingthe objects from the two sets. Using this approach, the greater set is the one inwhich some elements are left over. If nothing is left over in neither of the two sets,then the number of the two sets is the same, although we do not know what thenumber is. A mathematician would say that two sets have the same number ofelements when a bijection exists between them. If we use this approach for the setof natural numbers, strange things start happening. It is not very well known thatone of the �rst people arriving to this conclusion was Hugo Clever, who worked asa receptionist in Brno Grandhotel and faithful to his name, it had at that time anin�nite number of rooms. One day, when there was a downpour of rain outside,a tired old man came to the hotel and asked for a room. Mister Clever felt sorryfor this man and although the hotel was full up, he found a room for this guestthrough an organisational arrangement. We already suspect that he moved theguest from room number 1 to room number 2, guest from room number 2 to roomnumber 3, etc., and through this operation, room number 1 was free and no guestwas left without a room. Mister Mazany did it for free for the tired old man, butin other cases, he only did it for a bribe, through which he earned considerablemoney, especially in the times of fairs and Grand Prix competitions, because therumour about the clever receptionist who can accommodate n guests even whenthe hotel is full up spread fast. When Mister Clever managed to accommodate, foran adequate bribe, even a train with in�nitely many tourists (he moved each guestto the room whose number was a double of their original room, through which heobtained in�nitely many rooms with odd numbers), he stopped this activity anddecided to spend his money in the most popular holiday resorts.

Cantor was concerned with the problem of uniqueness of Fourier series, andhe found out that the condition can be weakened and in�nitely many exceptions,in some sense, may be allowed, without any detrimental e�ects to the uniqueness.Cantor therefore introduced (intuitively) the notion of set, one-to-one mappingbetween sets, and the notion cardinality. He stated that the set of natural numbersis countable and he denoted its cardinality as ℵ0. He proved that the set of rationalnumbers was countable; similarly, that the set of integers is also countable and on

6.3. THE THIRD CRISIS 59

the other hand, that also the set of squares of natural numbers is countable. As notevery natural number is a square of another natural number and since at the sametime each square of a natural number is a natural number, the two sets are in therelation of inclusion. On the other hand, there is a one-to-one mapping betweenthem, and they thus have the same cardinality (same number of elements). Cantorthus denied Euclid's common axiom 8, which states that the whole is larger thanits part. Cantor also proved that the set of real numbers is not countable, andthat there are thus at least two di�erent in�nities. Cardinality of the set of realnumbers is called continuum and is denoted by ℵ. For real numbers, it also holdsthat a whole need not be greater than its part, as the cardinality of any intervalof real numbers is again continuum. Finding a one-to-one mapping between twointervals is no problem at all.

At the same time, contemporary with the rise of the new theory, di�cultiesbegan to arise, or rather problems that could not be solved and that began to becalled paradoxes or antinomies of set theory. The �rst of these antinomies, knownalready to Cantor and published by Cesare Burali-Forti in 1897, can be formulatedin the following way: Ordinal number of a well-ordered set of all ordinal numbersis larger than all ordinal numbers (an ordinal number larger than itself exists).Another antinomy can be formulated in the following way: Let M be a set of allsets that are not their own elements. Then, however, it holds both that M ∈ Mand M /∈ M . Jules Richard (1862-1956) drew attention to another paradox in1905. His paradox consists in this: Any natural number can be described by asentence in some language. If we choose Czech, then that sentence is representedby a certain sequence of letters of the Czech alphabet. There are less than 50 lettersof Czech alphabet, and thus there are less than 50100 Czech sentences containingless than 100 letters. There are, however, more natural numbers than this numberand there is the least one of those, which we will denote by nm; nm is the leastnumber that cannot be described by a sentence with less than 100 letters. Butthat is exactly what we have done now.

There are, however, similar problems in logic. Epamenides was de�nitely con-vinced that he was the only truth-speaking Cretan. But according to his statement,all Cretans are liars, and thus he is also a liar. In that case it is not true thatall Cretans are liar. But if the one truth-speaking is Epamenides, then he uttereda truthful expression. The author also likes to remember the time when he as acute kid passed the Sunday afternoons in barbers' shop, because all man had tobe shaved on Sunday. It can thus be said that there were two kinds of men, in the�rst one, there were those who shaved themselves, and in the second one, thosewho were shaved by the barber. However, which group does barber belong to?

Mathematicians (and also philosophers) were looking for a way out of the crisis,but they did not act as a unanimous group and several directions of dealing withthe crisis emerged. The intuitionists rejected the actual in�nity and favoured in-tuition. Among other things, they rejected the principle of tertium non datur (foreach proposition, either V or ¬V holds). Luitzen Egbertus Brouwer (1881-1966)was the founder of intuitionism. As was suggested above, some antinomies from settheory have their analogies in logic. Logicism regarded logic to be the foundationof mathematics, and of course only the modern logic, without impredicativeness


(a Cretan cannot speak about Cretans). Bertrand Russell (1872-1970) and AlfredNorth Whitehead (1861-1947) are the two most important representatives of thisdirection. Formalism is best represented by David Hilbert (1862-1943), who re-fused to be thrown out of the paradise created for mathematics by Cantor. Themain idea of this direction is the establishment of a rigorous formal system like theone of Euclid, i.e. to establish a complete and consistent system of axioms, fromwhich any proposition could be proved with the aid of de�nitions. This mathe-matics was taught at schools and it seems that no other direction is possible anymore. However, then Kurt Gödel should not have been born in Brno, since hestated and proved the following proposition: Let S be a complete and consistentsystem of axioms. Then an undecidable proposition exists in this system.

As contemporary mathematicians agree, we have not overcome the third crisisyet. Each of the mentioned directions (and also those that have not been mentionedhere) certainly contributed to the development of mathematics, but it did not solvethe crisis. Is the situation in mathematics similar to the situation in physics at theend of 19th century when it seemed that after a couple of last problems are solved(photoelectric phenomenon, radiation of an absolutely black object), the physicistswould no longer have any work to do? Will a mathematical Einstein come and tearcontemporary mathematics to thrash? But just like Newtonian physics did notcease to exist and continues to serve well, nor will the mathematics of Pythagoras,Euclid, Archimedes, Fermat, Newton, Euler and other giants mentioned in theprevious sections cease to exist. Whether or not mathematicians come up withanything in future, the author thinks that the bricklayers will keep demarcating theright angle with the aid of the triangle with sides equal to 3, 4, and 5. Mathematicswill certainly not lose it position and anyone who deals with it should know itshistory, to which this book strives to contribute.

Part II

Biographies

61

63

In this part, we will introduce short biographies of a few important mathe-maticians whose names we encounter when teaching mathematics and elementaryschools and high schools. The selection is necessarily subjective, but the lengthof this text does not allow us to introduce all the people who contributed to theshape of mathematics taught in schools today. For scientists from the antiquityespecially, we only have a few biographical data that could be veri�ed in othersources, and therefore we sometimes mention facts that are perhaps not true, butonly good fables, because we think that even such things belong to teaching, sincethey help mathematics look more human.

64

Chapter 7

Foreign mathematicians

7.1 Archimedes of Syracuse

Archimedes was regarded the most signi�cant scientist of the Antiquity and wethink that this was well deserved. This personality of Hellenic science namelyachieved the thing that is demanded so often today, namely connecting theoryand practice. Using today's terminology, we could say that Archimedes excelledin both pure and applied research and that he mastered several branches of science.

He was born in 280 BC in Syracuse in Sicily. This town, in contrast to what itis today, was then one of the most important centres of Ancient Greek civilization,both in the classical and Hellenic period. In the times of Archimedes, the centreof education was in Alexandria. Archimedes also maintained scienti�c ties withhis colleagues in Mouseion.

As has already been said, he could connect theory and practice. His mostfamous discovery is the one about buoyancy forces exerted on an object partiallysubmerged in �uid; this principle is named after him and can be formulated asfollows: Any object immersed in a �uid, is buoyed up by a force equal to the weightof the �uid displaced by the submerged part of the object. This discovery is usuallyconnected with the task Archimedes was given by the tyrant king Hiero II. Thisking had a new gold crown made and he suspected the goldsmith of cheating.Therefore he asked Archimedes to solve this problem. However, Archimedes couldnot solve the problem, until he in a bath realised that he could solve the problemby weighing the crown in water and on dry land and compare the results with theweight of gold of the same weight. According to the legend, he was so enthusiasticabout his discovery that he ran out of the bath and crying heureka (I found it),ran to announce the solution to his king.

When Hannibal invaded Italy, the town of Syracuse joined him. In 215 BC,they were besieged by the Roman general Marcellus. Although the Romans byfar outnumbered them, the town of Syracuse resisted the siege for three years andapart from the bravery of the inhabitants, it was also thanks to Archimedes. Heemployed his knowledge fully in the service for the Syracuse army. He constructedcranes that could catch and turn over the Roman ships and the legend has it that

65

66 CHAPTER 7. FOREIGN MATHEMATICIANS

he could also put those ships on �re. Here, various authors disagree whether itis a fact, or a myth. If we stick with the �rst version, then Archimedes probablyused the fact that the shields of soldiers are in fact mirrors. Then all it took toput the ships on �re was to set those "mirrors" up in such a way that their focimet on a Roman ship, where the solar energy was thus also concerned.

In 212, the town of Syracuse lost and in connection with the invasion of thevictorious army into the city, the life of Archimedes came to its end as well.Although Marcellus issued the command that the life of this genius should besaved, probably in the hope that he could serve Rome, that was not to be and thelife of Archimedes ended when his town of birth lost its freedom. The legend saysthat exactly in the moment when the invaders entered the house of Archimedes, hewas engaged in a problem concerning circles while he drew those circle in the sand.Roman soldiers took his request not to disturb his circles as a sign of resistance andone of them pierced Archimedes with his spear. Sources do not provide the nameof this soldier and the name of his commander is only known to those well versedin the history of Rome. However, the name of Archimedes was not forgotten.Apart from his principle, we also know the screw of Archimedes and the spiral ofArchimedes.

7.2 Lazare Carnot

In mathematics for elementary or high schools, we do not meet the name Carnot.We have nevertheless decided to include him in this selection. We will talk aboutthe reasons for this later. Lazare Nicolas Marguerite Carnot was born on May3, 1753 in a middle-class family in the town of Nolay in Burgundy. Since hisyouth, he demonstrated great talent for mathematics and technology, and thus itis not surprising that he graduated from the prestigious school in Mezieres, whereGaspard Monge also taught. After graduation in 1773, he became an o�cer withthe French army, he served with various garrisons and he later became a captainthe royal army.

When Bastille fell on July 14, 1789, he put all his abilities in the service of therevolution. Politically, he was close to the Jacobites, and besides, he had knownMaxmilien Robespierre and this charismatic man had great in�uence on Carnot'sopinions. Carnot's big moments came in 1793 when he became a member of thecommittee of public safety and was responsible for the army. During a short periodof time, he performed a reform of the army and the French revolutionary forcesdefeated the intervention armies of Austria and Prussia. Among other things,he introduced universal conscription for young men from the age of 18 to 25, heeliminated noblemen, but also the illiterate, from the positions of o�cers andhe also joined the voluntary units with regular army. We cannot forget also thefounding of typography o�ce from which military headquarters was later formed.He more than deserved the honorary title the Organizer of the Victory, with whichhe was decorated.

Carnot did not change his conviction, he especially protested against absolutisttendencies and non-democratic methods, and thus he had to �ee into emigration

7.3. RENÉ DESCARTES 67

several times. Although his opinions were close to those of Robespierre, his meth-ods of government were alien to him and therefore he aided his defeat. For similarreasons, his relationship with Napoleon was rather precarious for the same reason,although they respected each other. Napoleon appreciated his abilities by promot-ing him to the position of division general and awarded him the title of an earl.Carnot then did not hesitate to support Napoleon upon his return from Elba andaccepted the position of minister of the interior. The Bourbons did not forgivehim his service during the revolution, so after their restoration, he had to leaveFrance for good. He lived in Magdeburg for the rest of his life, where he worked asa scientist and also an advisor of the Prussian government for the issues of militaryschools and fortress construction. He died on August 2, 1823 and was buried inBerlin. In 1889, the French government decided that his remains be transportedto France and stored in Pantheon.

Carnot is one of the most important mathematicians of his time. His �eld ofinterest was mainly geometry, e.g. Carnot's theorem. In his time, he was also anexpert on fortress construction.

The name "Carnot" should be known to any high school student who stud-ied physics, or rather thermodynamics. However, Lazare's share in this discoveryis only indirect. Carnot's cycle and other discoveries are due to his son, Nico-las Léonard Sadi Carnot (1796-1832), who inherited father's genes for science.In 1887, Marie Francois Sadi Carnot (1838-1894), the great-grandnephew of theOrganizer of Victory, was elected French president.

7.3 René Descartes

René Descartes was born on March 31, 1596 in la Haye in France. He was educatedin a Jesuit college La Fleche in Anjou, where he was sent as an eight-year old. Inhis youth, he travelled a lot and also served in the army. He was in the army ofMaxmilian of Bavaria, �ghting against Bohemian nobility, but there is no evidencefor his participation at the battle of the White Mountain. In 1628, he settled inHolland, where he lived for more than 20 years. In 1649, he could not resist theinvitation of the outstanding woman on the Swedish throne, Queen Christina, andwent to see her in Stockholm. The tough northern weather, however, was not goodfor his health and he died of pneumonia on February 11, 1650.

Descartes belonged to the greatest scientists of his time. In contrast to hisscienti�c colleague and rival Fermat, he was not afraid of publishing. In 1637, hepublished his Discours de la méthod pour bien conduire sa raison et chercher lavérité dans les science. This book contained three supplements, from which themost important is the last one of those, entitled La Géometrie. It was in thissupplement that he published the elements of analytical geometry, i.e. the partof mathematics that solves geometrical problems in an algebraic way. He alsosucceeded in simplifying mathematical symbolism, denoting the unknowns by theletters from the end of the alphabet and the constants by the letters from thebeginning of the alphabet comes from him. We also have Cartesian coordinatesystem, but here, the primacy belongs to Pierre de Fermat. Descartes also solved


problems in physics, especially in mechanics and optics, and he a.o. gave a fulltheory of the rainbow.

Descartes is naturally very well-known also to those interested in the humani-ties, because he is one of the best known philosophers and he also devoted time topsychology. Maybe it was success in mathematics that led Descartes to introducesimilar methods also in philosophy; this direction is called rationalism. Its basisis his famous statement I know, therefore I am. He distinguishes between Godas the uncreated and in�nite substance and two created substances - the worldof the body and the world of the mind with the attribute of thinking. He triedto formulate a general method of recognition, based on four principles: 1) acceptonly those things that are clear and apparent and without any doubts; 2) divideevery problem into simpler parts that can be recognized easily; 3) proceed fromthe simple to the di�cult; 4) put together complete lists and general overviews,so that it would become clear that we have not forgotten anything.

7.4 Leonhard Euler

Leonhard Euler was born on April 15, 1707 in Basel. His father was an evangelicpriest and wanted his son to follow him. Luckily, this did not happen and theworld did not lose one of its greatest mathematicians of all time. Johan Bernoulli,his teacher at the university in Basel, is to be credited with this. Although he wasSwiss by birth, described in sports terminology, he played for Russia and Prussia.The emperors of these countries namely realized the importance of science andtherefore they could draw the best scientists to the prestigious workplaces.

Apart from talent for mathematics, Euler also had phenomenal memory andimagination, and thus he could work until the end of his life, although he lostsight. His productivity was admirable: thanks to his manuscripts, the printers inSt Petersburg had work even dozens of years after his death. However, he was farfrom the stereotype of a mathematician as an absent-minded man who lives onlyfor his science and is not �t for normal life. It is said that he was a jovial manwho lived a normal life.

As he touched upon all areas of mathematics with his work, it is very di�cultto assess his work on such a small space. His work contributed to the developmentof number theory. In his work, he basically went in Fermat's footsteps and hesucceeded in proving or disproving most of his statements; he for example provedthe Last Theorem of Fermat for n = 3. He extended or generalized many ofFermat's �ndings, for example, he extended Fermat's little theorem for compositemodules. His solution of the problem of seven bridges of Königsberg became afoundational stone of a new discipline - graph theory [Sa].

The core of his work, however, lies in mathematical analysis. His de�nition ofthe function z from 1755 extends the notion of a function from a mere analyticalexpression to a mutual dependence of two quantities. When some quantities dependon others in such a way that when we change them (the second ones), they alsochange, we say that the �rst ones are functions of the second ones. This name hasan outstandingly broad character and includes all possible ways of expressing one

7.5. PIERRE DE FERMAT 69

quantity through other quantities.

7.5 Pierre de Fermat

Toulose judge Pierre de Fermat belongs to the most notable mathematicians inthe world. He is usually called the prince of the amateurs, but in his times,mathematics as a profession did not exist, and therefore this denotation is slightlymisleading. He was born in 1601, at least according to the publications untiltoday. Fermat's father was a rich merchant with leather and he gave his songood education of a lawyer. Pierre chose his distant relative Luisa de Long as hislifetime partner. Together, they had two sons and a daughter. More than twocenturies before the �rst aria of Kecal was performed, Fermat also counted whathe could gain. Thanks to his good �nancial situation, he could buy a judge o�cein Toulouse, where he spent all his life. He divided his time between the o�ce,family, and his great hobby, mathematics.

It is not known exactly when he started to devote time to mathematics, but itis certain that roughly in the 1630s, he joined the circle around Father Mersenne.In the time of Fermat, there were no scienti�c journals and no scienti�c conferenceswere organized. Scientists therefore had to publish their �ndings in book, whichFermat, for various reasons, rejected. There was a further option, correspondencewith colleagues, and Mersenne improved this possibility through becoming a sortof a scienti�c secretary of a group of scientists and an organizer of a permanent cor-responding scienti�c conference. For Fermat, this exchange was very convenient,because it allowed him to stay detached from the world.

A saying tells us that each man should bring up a son and for Fermat, ful�llingthis obligation was a key one. This man, above all, paid attention to his job andscience was only a hobby for him. Although he corresponded with the Europeanscienti�c elite of that time, he did not take extra care of keeping his correspondence.His elder son Samuel succeeded his father in the o�ce of the judge and he was alsointerested in mathematics, but did not inherit his father's talent. Nevertheless, heis to be credited with collecting the preserved letters from his father's inheritanceas well as from his correspondents and he published these letters. If it was notfor Samuel Fermat, the name of Fermat would not be known to historians ofmathematics, or perhaps it would be included only in the most detailed books.Mathematics would also be deprived of one of its greatest mysteries, called inCzech Fermat's great theorem (in English known as Fermat's last theorem).

Fermat also had at his disposal Bachet's edition of the Arithmetics by Diophan-tus and this book interested him so much that he not only studied it in detail,but also wrote his notes and commentaries directly in this book. One problemsolved by Diophantus concerned Pythagorean triples and Fermat wrote the fol-lowing remark: It is impossible to divide a cube into two cubes, biquadrate intotwo biquadrates, and in general any power into two powers. I found a beautifulproof for this fact, but this margin is too narrow. In our contemporary mathemat-ical language, the Diophantine equation xn + yn = zn does not have a solution inintegers for n > 2. The simple formulation and Fermat's bold statement that he


has a proof was an enormous challenge and since its publication until these days,both professionals and amateurs tried to �nd the lost proof. Wiles did indeedprove this hypothesis in 1994, but he used such methods that could never havebeen available to Fermat. Thus even after the proof has been published, attemptsto �nd Fermat's beautiful proof did not cease and Fermat's last theorem has be-come the same thing for mathematics as the perpetuum mobile for the physicists.Today, most researchers are inclined to believe that Fermat incorrectly generalizedhis �ndings about certain concrete values and that this theorem cannot be provedin elementary mathematics.

Fermat is considered the founding father of number theory. Apart from Fer-mat's last theorem which has just been mentioned, we also know Fermat's littletheorem, ap−1 ≡ 1 mod p, where p is a prime and (a, p) = 1. He also paidattention to numbers of the form Fn = 22n + 1 (Fermat numbers), about whichhe mistakenly assumed that they are primes, but he did not have a proof for this.This problem has not been solved until today, but so far no primes were found,except for the �rst �ve that were known to Fermat already; on the contrary manyof those numbers were factorized [Le].

Fermat belonged to the greatest �gures of world science in the �rst half of the17th century. He laid the foundations of analytical geometry and only because ofthe fact that Descartes published his ideas in a book, while Fermat's ideas wereonly known to a narrow circle of people, we now have Cartesian coordinate system,and not Fermat's. His correspondence with Blaise Pascal laid the foundationalstone of probability theory. He knew the quadrature of the parabola, he couldwrite the equation determining the tangent to a curve, and he is the founder ofmodern number theory. He could solve problems in mechanics and optics and wecan �nd Fermat's principle, describing the spread of light, in high school textbooksfor physics.

Fermat was a source of inspiration for mathematicians. It may seem that afterhis famous hypothesis has been solved, there is nothing that can surprise us aboutthis Frenchman. However, towards the end of 20th century, a German historian ofmathematics realized that the age given on Fermat's tomb does not comply withthe dates given in literature and con�rmed by registers. As we cannot assume thatSamuel did not know his father's age, here is a possible explanation: Pierre bornin 1601 died as a small boy, and at the same time, his father became a widower.He married for the second time and had a son with his second wife, whose namewas again Pierre. The registers for this period are lost, so we cannot prove thishypothesis.

7.6 Johann Carl Friedrich Gauss

Gauss was born on April 30, 1777 in Braunschweig as the only son of a bricklayerand a water master. He was very gifted and if he attended school these days, hewould probably be labelled as hyperactive by the psychologists. It is said thatthe teacher gave him the task of adding up numbers from 1 to 1000 in order tokeep him busy for at least some short period of time. What a surprise it was for

7.6. JOHANN CARL FRIEDRICH GAUSS 71

the teacher when the little Johann announced the correct result 500 500 after ashort while. He namely noticed that the sum of the �rst and the last number isthe same as the sum of the second and penultimate number and so on. It thussu�ces to add the �rst and the last number and multiply this sum by the numberof such pairs, or, if you like, to use the formula for calculating the sum of the �rstn members of an arithmetic formula sn = a1+an

2 ·n. This story is well known, butit is less well known that as a three-year old, Gauss drew his father's attention tothe mistake in the calculation of wages for the workers.

We can also meet the name Gauss in mechanics, electrostatics, magnetism, andastronomy, but we will focus on his contribution to mathematics. When complexnumbers are taught, everybody will probably think of Gaussian complex plane,or the one-to-one mapping of complex numbers and points in the place. Whenwe have experimental data and want to present them in a graphical way, we willuse the method of least squares. This method allows us to construct such a curvefrom the measured values that the sum of the squares of the di�erences was thesmallest possible. He also dealt with the construction of regular polygons andproved that Euclidean construction is only possible for n = 2kF1F2 · · ·Fn, whereFi are Fermat's primes (see the section on Fermat). Therefore it is possible toconstruct a regular 17-gon, whose construction Gauss found, but not a regularheptagon, although 7 is regarded to be a lucky number.

His Disquisitiones artihmetiquae belongs to key works in number theory. Healso introduced the notion of congruence and proved the law of quadratic reci-procity. He also investigated the �fth postulate and concluded that apart from Eu-clidean geometry, other geometries may exist, but he did not publish his thoughtsin this direction. His fruitful life came to an end on February 23, 1855 in Göttingen,where had taught at the university for many years.


Chapter 8

Czech mathematicians

8.1 Mathias Lerch

Mathias Lerch was born in Milínov by Su²ice on February 1860 in a family ofa minor farmer. His life was in�uence by an accident (fall from the loft), whenhe seriously hurt his leg, which remained bent in the knee, and thus the youngMathias could only walk with crutches. He only started attending school whenhe was 9 years old when his family moved to Su²ice. Already at elementaryschool, the young Mathias demonstrated his talents, so it is no wonder that hedecided to study �rst a secondary school and then the Czech Technical Universityin Prague. The consequences of his injury, however, haunted him all the time.Given his physical de�ciency, he could not become a teacher at a grammar school,as he had wanted. He therefore decided for the career of a mathematician and auniversity teacher, which was very lucky for Czech mathematics. As a grammarschool teacher, he probably would have been only mediocre, but in mathematics,he was very successful. After studying at Charles University and then for one yearin Berlin, he became assistant at Czech Technical University in Prague. At thattime, his publication activity started and he published in both Czech and foreignjournals.

In 1895, another signi�cant change happened in his life. As he could not goon being an assistant, he tried to obtain professorship at a university in Czechlands. His e�orts, however, were not successful, which was strange, because Lerchwas number one among the living Czech mathematician at that time already. Itis possible that envy played a role therein and also the fact that Lerch adequatelyboasted about his qualities. Luckily, an o�er from the university in the SwissFribourg appeared which Lerch accepted and was appointed professor at thatuniversity. The ten years that he had passed in Switzerland were probably thehappiest years in Lerch's life, both in the professional and the personal one. In1900, he went through a di�cult operation which was nevertheless successful andLerch could exchange the crutches for a walking stick. In that year, his nieceR·ºena Sejpková came to visit him (Lerch married her in 1921) and started takingcare of his household matters. At that time, also Lerch's publishing activities

73

74 CHAPTER 8. CZECH MATHEMATICIANS

culminated and his work was awarded the prestigious Great Prize of the ParisAcademy of the Sciences in 1900. Lerch was the �rst Czech mathematician toreceive this prize and a second Czech (after Purkyn¥).

Despite all his success, Lerch longed for return to his home country. He thenstayed in Brno, as he became the professor of mathematics at the Czech technicaluniversity in Brno in 1906. When the university was founded in 1919, it was rathernatural that he became the �rst professor of mathematics there and that he thus�nally had a position that was adequate to his qualities. However, he did notenjoy this position for long, since he got a cold after swimming in the river anddied of pneumonia in Su²ice on August 3, 1922.

Lerch was unquestionably an outstanding mathematician, but his personalitywas rather controversial. Maybe it was a consequence of his childhood injuryand his bad health condition in later years, because he su�ered from diabetes, atthat time an incurable illness. In his action, he was direct and would support hisviews even in spite of con�icts. Already in his youth, he had to go from Pilsen toRakovník after his con�ict with a catechist. His di�culties with �nding a positionhave already been mentioned. On the other hand, he was thought of highly andwas taken seriously.

Lerch was strongly in�uenced by Kronecker and his work was in the style ofthis German mathematician. He solved concrete problems, although he publishedmore than two hundred papers, he never wrote a monograph, although his qual-ities in some areas of mathematics (elliptic integrals, malmsten series, quadraticforms) were outstanding and he would de�nitely be able to write the monograph.Although he was not a good teacher as far as the form is concerned (feeble andmonotonous voice etc.), his lectures were excellent from the point of view of con-tent. He allegedly said that when he had not been good for Czech schools, theywere then not good for him and did not deserve his monographs. Despite that,Lerch has put his mark onto Czech schools and especially in the 1970, when settheory became the basis of teaching, he was the indirect nightmare of the parentsof the schoolchildren, as it is almost certain that he is the author of the Czechword for set.

Bibliography

[Ba] Balada F.: Z d¥jin elementární matematiky. Státní pedagogickénakladatelství Praha 1959

[Be] Beckmann, P.: Historie £ísla pí. Academia Praha 1998 (p°eloºilLibor Pátý.)

[BF] Historie matematiky I. Sborník ze Seminá°e pro vyu£ující nast°edních ²kolách v Jeví£ku, srpen 1993. Edito°i Be£vá° J., FuchsE. J�MF, Brno 1994

[Be2] Be£vá° J. a kol.: Matematika ve st°edov¥ké Evrop¥. PrometheusPraha 2001

[Eu] http://aleph0.clarku.edu/ djoyce/java/elements/elements.html

[Ja] Jaro²ová M.: Souvislost Fibonacciho £ísel s jinými matemat-ickými pojmy. In Matematika v prom¥nách v¥k· IV, editor. E.Fuchs. Akademické nakladatelství Cerm, Brno 2007

[Le] Lepka K.: Historie Fermatových kvocient· (Fermat�Lerch).Edice d¥jiny matematiky, sv. 14 Prometheus Praha 2000

[Ma1] Ma£ák K.: T°i st°edov¥ké sbírky matematických úloh.Prometheus Praha 2001

[Ma2] Ma£ák K.: Po£átky po£tu pravd¥podobnosti. Edice D¥jinymatematiky, sv. 9. Prometheus Praha 1997

[Si] Singh S.: Velká Fermatova v¥ta. Academia Praha 2000

[Sm1] �imerka V.: Algebra £ili po£tá°ství obecné pro vy²²í gymnasia:tiskem a nákladem Dr. E. Grégra 1863

[Sm2] �imerka V.: Síla p°esv¥d£ení. �asopis pro p¥stování mathe-matiky a fysiky 11(1881), 75�111

[Sa] �i²ma P.: Teorie graf· 1736�1963. Edice D¥jiny matematiky, sv.8. Prometheus Praha 1997

75

76 BIBLIOGRAPHY

[Sch] Schwabik �., �armanová P.: Malý pr·vodce historií integrálu.Prometheus Praha 1996

[Vy] Vymazalová H.: Staroegyptská matematika. Hieratické matem-atické texty. �eský egyptologický ústav Praha 2006

[Zn] Znám �. a kol.: Pohl�ad do d¥jín matematiky. Alfa Bratislava1986

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